We are currently solving the following issue within our repository. Here is the issue text:
--- BEGIN ISSUE ---
AbelianGroupElement does not always reduce exponent vectors when needed
In SageMath 9.8:
```
sage: A = AdditiveAbelianGroup([3])
sage: A([6]) == A([0])
True
sage: A([6]).vector()
(0)
sage: M = AbelianGroup([3])
sage: M([6]) == M([0])  # output should be True
False
sage: M([6]).exponents()
(6,)
```
The last output contradicts the documentation of the `exponents()` method, which says
```
   OUTPUT:

   A tuple of integers for an abelian group element. The integer can
   be arbitrary if the corresponding generator has infinite order. If
   the generator is of finite order, the integer is in the range from
   0 (inclusive) to the order (exclusive).
```
The `exponents()` method should be fixed (so that it reduces the exponents modulo the respective orders) and `_richcmp_()` should use this method.  (Alternatively, `_richcmp_()` and possibly other methods could do the reduction modulo the orders, but this is probably be more fragile.)

--- END ISSUE ---

Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
--- BEGIN FILES ---
Path: `src/sage/groups/abelian_gps/element_base.py`
Content:
```
1 """
2 Base class for abelian group elements
3 
4 This is the base class for both
5 :mod:`~sage.groups.abelian_gps.abelian_group_element` and
6 :mod:`~sage.groups.abelian_gps.dual_abelian_group_element`.
7 
8 As always, elements are immutable once constructed.
9 """
10 
11 
12 ###########################################################################
13 #  Copyright (C) 2006 William Stein <wstein@gmail.com>
14 #  Copyright (C) 2006 David Joyner  <wdjoyner@gmail.com>
15 #  Copyright (C) 2012 Volker Braun  <vbraun.name@gmail.com>
16 #
17 #  Distributed under the terms of the GNU General Public License (GPL)
18 #                  https://www.gnu.org/licenses/
19 ###########################################################################
20 
21 from sage.structure.element import MultiplicativeGroupElement
22 from sage.misc.cachefunc import cached_method
23 from sage.arith.all import GCD, LCM
24 from sage.rings.integer import Integer
25 from sage.rings.integer_ring import ZZ
26 from sage.rings.infinity import infinity
27 from sage.structure.richcmp import richcmp
28 
29 
30 class AbelianGroupElementBase(MultiplicativeGroupElement):
31     """
32     Base class for abelian group elements
33 
34     The group element is defined by a tuple whose ``i``-th entry is an
35     integer in the range from 0 (inclusively) to ``G.gen(i).order()``
36     (exclusively) if the `i`-th generator is of finite order, and an
37     arbitrary integer if the `i`-th generator is of infinite order.
38 
39     INPUT:
40 
41     - ``exponents`` -- ``1`` or a list/tuple/iterable of integers. The
42       exponent vector (with respect to the parent generators) defining
43       the group element.
44 
45     - ``parent`` -- Abelian group. The parent of the group element.
46 
47     EXAMPLES::
48 
49         sage: F = AbelianGroup(3,[7,8,9])
50         sage: Fd = F.dual_group(names="ABC")
51         sage: A,B,C = Fd.gens()
52         sage: A*B^-1 in Fd
53         True
54     """
55 
56     def __init__(self, parent, exponents):
57         """
58         Create an element.
59 
60         EXAMPLES::
61 
62             sage: F = AbelianGroup(3,[7,8,9])
63             sage: Fd = F.dual_group(names="ABC")
64             sage: A,B,C = Fd.gens()
65             sage: A*B^-1 in Fd
66             True
67         """
68         MultiplicativeGroupElement.__init__(self, parent)
69         n = parent.ngens()
70         if exponents == 1:
71             self._exponents = tuple( ZZ.zero() for i in range(n) )
72         else:
73             self._exponents = tuple( ZZ(e) for e in exponents )
74             if len(self._exponents) != n:
75                 raise IndexError('argument length (= %s) must be %s.'%(len(exponents), n))
76 
77     def __hash__(self):
78         r"""
79         TESTS::
80 
81             sage: F = AbelianGroup(3,[7,8,9])
82             sage: hash(F.an_element()) # random
83             1024
84         """
85         return hash(self.parent()) ^ hash(self._exponents)
86 
87     def exponents(self):
88         """
89         The exponents of the generators defining the group element.
90 
91         OUTPUT:
92 
93         A tuple of integers for an abelian group element. The integer
94         can be arbitrary if the corresponding generator has infinite
95         order. If the generator is of finite order, the integer is in
96         the range from 0 (inclusive) to the order (exclusive).
97 
98         EXAMPLES::
99 
100             sage: F.<a,b,c,f> = AbelianGroup([7,8,9,0])
101             sage: (a^3*b^2*c).exponents()
102             (3, 2, 1, 0)
103             sage: F([3, 2, 1, 0])
104             a^3*b^2*c
105             sage: (c^42).exponents()
106             (0, 0, 6, 0)
107             sage: (f^42).exponents()
108             (0, 0, 0, 42)
109         """
110         return self._exponents
111 
112     def _libgap_(self):
113         r"""
114         TESTS::
115 
116             sage: F.<a,b,c> = AbelianGroup([7,8,9])
117             sage: libgap(a**2 * c) * libgap(b * c**2)
118             f1^2*f2*f6
119         """
120         from sage.misc.misc_c import prod
121         from sage.libs.gap.libgap import libgap
122         G = libgap(self.parent())
123         return prod(g**i for g,i in zip(G.GeneratorsOfGroup(), self._exponents))
124 
125     def list(self):
126         """
127         Return a copy of the exponent vector.
128 
129         Use :meth:`exponents` instead.
130 
131         OUTPUT:
132 
133         The underlying coordinates used to represent this element.  If
134         this is a word in an abelian group on `n` generators, then
135         this is a list of nonnegative integers of length `n`.
136 
137         EXAMPLES::
138 
139             sage: F = AbelianGroup(5,[2, 3, 5, 7, 8], names="abcde")
140             sage: a,b,c,d,e = F.gens()
141             sage: Ad = F.dual_group(names="ABCDE")
142             sage: A,B,C,D,E = Ad.gens()
143             sage: (A*B*C^2*D^20*E^65).exponents()
144             (1, 1, 2, 6, 1)
145             sage: X = A*B*C^2*D^2*E^-6
146             sage: X.exponents()
147             (1, 1, 2, 2, 2)
148         """
149         # to be deprecated (really, return a list??). Use exponents() instead.
150         return list(self._exponents)
151 
152     def _repr_(self):
153         """
154         Return a string representation of ``self``.
155 
156         OUTPUT:
157 
158         String.
159 
160         EXAMPLES::
161 
162             sage: G = AbelianGroup([2])
163             sage: G.gen(0)._repr_()
164             'f'
165             sage: G.one()._repr_()
166             '1'
167         """
168         s = ""
169         G = self.parent()
170         for v_i, x_i in zip(self.exponents(), G.variable_names()):
171             if v_i == 0:
172                 continue
173             if len(s) > 0:
174                 s += '*'
175             if v_i == 1:
176                 s += str(x_i)
177             else:
178                 s += str(x_i) + '^' + str(v_i)
179         if s:
180             return s
181         else:
182             return '1'
183 
184     def _richcmp_(self, other, op):
185         """
186         Compare ``self`` and ``other``.
187 
188         The comparison is based on the exponents.
189 
190         OUTPUT:
191 
192         boolean
193 
194         EXAMPLES::
195 
196             sage: G.<a,b> = AbelianGroup([2,3])
197             sage: a > b
198             True
199 
200             sage: Gd.<A,B> = G.dual_group()
201             sage: A > B
202             True
203         """
204         return richcmp(self._exponents, other._exponents, op)
205 
206     @cached_method
207     def order(self):
208         """
209         Return the order of this element.
210 
211         OUTPUT:
212 
213         An integer or ``infinity``.
214 
215         EXAMPLES::
216 
217             sage: F = AbelianGroup(3,[7,8,9])
218             sage: Fd = F.dual_group()
219             sage: A,B,C = Fd.gens()
220             sage: (B*C).order()
221             72
222 
223             sage: F = AbelianGroup(3,[7,8,9]); F
224             Multiplicative Abelian group isomorphic to C7 x C8 x C9
225             sage: F.gens()[2].order()
226             9
227             sage: a,b,c = F.gens()
228             sage: (b*c).order()
229             72
230             sage: G = AbelianGroup(3,[7,8,9])
231             sage: type((G.0 * G.1).order())==Integer
232             True
233         """
234         M = self.parent()
235         order = M.gens_orders()
236         L = self.exponents()
237         N = LCM([order[i]/GCD(order[i],L[i]) for i in range(len(order)) if L[i]!=0])
238         if N == 0:
239             return infinity
240         else:
241             return ZZ(N)
242 
243     multiplicative_order = order
244 
245     def _div_(left, right):
246         """
247         Divide ``left`` and ``right``
248 
249         TESTS::
250 
251             sage: G.<a,b> = AbelianGroup(2)
252             sage: a/b
253             a*b^-1
254             sage: a._div_(b)
255             a*b^-1
256         """
257         G = left.parent()
258         assert G is right.parent()
259         exponents = [ (x-y)%order if order!=0 else x-y
260                       for x, y, order in
261                       zip(left._exponents, right._exponents, G.gens_orders()) ]
262         return G.element_class(G, exponents)
263 
264     def _mul_(left, right):
265         """
266         Multiply ``left`` and ``right``
267 
268         TESTS::
269 
270             sage: G.<a,b> = AbelianGroup(2)
271             sage: a*b
272             a*b
273             sage: a._mul_(b)
274             a*b
275         """
276         G = left.parent()
277         assert G is right.parent()
278         exponents = [ (x+y)%order if order!=0 else x+y
279                       for x, y, order in
280                       zip(left._exponents, right._exponents, G.gens_orders()) ]
281         return G.element_class(G, exponents)
282 
283     def __pow__(self, n):
284         """
285         Exponentiate ``self``
286 
287         TESTS::
288 
289             sage: G.<a,b> = AbelianGroup(2)
290             sage: a^3
291             a^3
292         """
293         m = Integer(n)
294         if n != m:
295             raise TypeError('argument n (= '+str(n)+') must be an integer.')
296         G = self.parent()
297         exponents = [ (m*e) % order if order!=0 else m*e
298                       for e,order in zip(self._exponents, G.gens_orders()) ]
299         return G.element_class(G, exponents)
300 
301     def __invert__(self):
302         """
303         Return the inverse element.
304 
305         EXAMPLES::
306 
307             sage: G.<a,b> = AbelianGroup([0,5])
308             sage: a.inverse()  # indirect doctest
309             a^-1
310             sage: a.__invert__()
311             a^-1
312             sage: a^-1
313             a^-1
314             sage: ~a
315             a^-1
316             sage: (a*b).exponents()
317             (1, 1)
318             sage: (a*b).inverse().exponents()
319             (-1, 4)
320         """
321         G = self.parent()
322         exponents = [(-e) % order if order != 0 else -e
323                      for e, order in zip(self._exponents, G.gens_orders())]
324         return G.element_class(G, exponents)
325 
326     def is_trivial(self):
327         """
328         Test whether ``self`` is the trivial group element ``1``.
329 
330         OUTPUT:
331 
332         Boolean.
333 
334         EXAMPLES::
335 
336             sage: G.<a,b> = AbelianGroup([0,5])
337             sage: (a^5).is_trivial()
338             False
339             sage: (b^5).is_trivial()
340             True
341         """
342         return all(e==0 for e in self._exponents)
343
```
--- END FILES ---

Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.

Here is an example:

```diff
diff --git a/examples/server_async.py b/examples/server_async.py
--- a/examples/server_async.py
+++ b/examples/server_async.py
@@ -313,4 +313,4 @@
 
 
 if __name__ == "__main__":
-    asyncio.run(run_async_server("."), debug=True)
+    asyncio.run(run_async_server(), debug=True)
diff --git a/examples/server_sync.py b/examples/server_sync.py
--- a/examples/server_sync.py
+++ b/examples/server_sync.py
@@ -313,5 +313,5 @@
 
 
 if __name__ == "__main__":
-    server = run_sync_server(".")
+    server = run_sync_server()
     server.shutdown()

```

diff --git a/src/sage/groups/abelian_gps/element_base.py b/src/sage/groups/abelian_gps/element_base.py
--- a/src/sage/groups/abelian_gps/element_base.py
+++ b/src/sage/groups/abelian_gps/element_base.py
@@ -64,15 +64,25 @@
             sage: A,B,C = Fd.gens()
             sage: A*B^-1 in Fd
             True
+
+        Check that :issue:`35216` is fixed::
+
+            sage: M = AbelianGroup([3])
+            sage: M([5]) == M([2])
+            True
+            sage: M([3]).is_trivial()
+            True
         """
         MultiplicativeGroupElement.__init__(self, parent)
         n = parent.ngens()
         if exponents == 1:
-            self._exponents = tuple( ZZ.zero() for i in range(n) )
+            self._exponents = tuple(ZZ.zero() for i in range(n))
         else:
-            self._exponents = tuple( ZZ(e) for e in exponents )
-            if len(self._exponents) != n:
-                raise IndexError('argument length (= %s) must be %s.'%(len(exponents), n))
+            if len(exponents) != n:
+                raise IndexError('argument length (= %s) must be %s'
+                                 % (len(exponents), n))
+            self._exponents = tuple(ZZ(e % o if o else e) for e, o in
+                                    zip(exponents, parent.gens_orders()))
 
     def __hash__(self):
         r"""
@@ -256,9 +266,8 @@
         """
         G = left.parent()
         assert G is right.parent()
-        exponents = [ (x-y)%order if order!=0 else x-y
-                      for x, y, order in
-                      zip(left._exponents, right._exponents, G.gens_orders()) ]
+        exponents = [x - y for x, y in
+                     zip(left._exponents, right._exponents)]
         return G.element_class(G, exponents)
 
     def _mul_(left, right):
@@ -275,9 +284,8 @@
         """
         G = left.parent()
         assert G is right.parent()
-        exponents = [ (x+y)%order if order!=0 else x+y
-                      for x, y, order in
-                      zip(left._exponents, right._exponents, G.gens_orders()) ]
+        exponents = [x + y for x, y in
+                     zip(left._exponents, right._exponents)]
         return G.element_class(G, exponents)
 
     def __pow__(self, n):
@@ -294,8 +302,7 @@
         if n != m:
             raise TypeError('argument n (= '+str(n)+') must be an integer.')
         G = self.parent()
-        exponents = [ (m*e) % order if order!=0 else m*e
-                      for e,order in zip(self._exponents, G.gens_orders()) ]
+        exponents = [m * e for e in self._exponents]
         return G.element_class(G, exponents)
 
     def __invert__(self):
@@ -319,8 +326,7 @@
             (-1, 4)
         """
         G = self.parent()
-        exponents = [(-e) % order if order != 0 else -e
-                     for e, order in zip(self._exponents, G.gens_orders())]
+        exponents = [-e for e in self._exponents]
         return G.element_class(G, exponents)
 
     def is_trivial(self):

{"golden_diff": "diff --git a/src/sage/groups/abelian_gps/element_base.py b/src/sage/groups/abelian_gps/element_base.py\n--- a/src/sage/groups/abelian_gps/element_base.py\n+++ b/src/sage/groups/abelian_gps/element_base.py\n@@ -64,15 +64,25 @@\n             sage: A,B,C = Fd.gens()\n             sage: A*B^-1 in Fd\n             True\n+\n+        Check that :issue:`35216` is fixed::\n+\n+            sage: M = AbelianGroup([3])\n+            sage: M([5]) == M([2])\n+            True\n+            sage: M([3]).is_trivial()\n+            True\n         \"\"\"\n         MultiplicativeGroupElement.__init__(self, parent)\n         n = parent.ngens()\n         if exponents == 1:\n-            self._exponents = tuple( ZZ.zero() for i in range(n) )\n+            self._exponents = tuple(ZZ.zero() for i in range(n))\n         else:\n-            self._exponents = tuple( ZZ(e) for e in exponents )\n-            if len(self._exponents) != n:\n-                raise IndexError('argument length (= %s) must be %s.'%(len(exponents), n))\n+            if len(exponents) != n:\n+                raise IndexError('argument length (= %s) must be %s'\n+                                 % (len(exponents), n))\n+            self._exponents = tuple(ZZ(e % o if o else e) for e, o in\n+                                    zip(exponents, parent.gens_orders()))\n \n     def __hash__(self):\n         r\"\"\"\n@@ -256,9 +266,8 @@\n         \"\"\"\n         G = left.parent()\n         assert G is right.parent()\n-        exponents = [ (x-y)%order if order!=0 else x-y\n-                      for x, y, order in\n-                      zip(left._exponents, right._exponents, G.gens_orders()) ]\n+        exponents = [x - y for x, y in\n+                     zip(left._exponents, right._exponents)]\n         return G.element_class(G, exponents)\n \n     def _mul_(left, right):\n@@ -275,9 +284,8 @@\n         \"\"\"\n         G = left.parent()\n         assert G is right.parent()\n-        exponents = [ (x+y)%order if order!=0 else x+y\n-                      for x, y, order in\n-                      zip(left._exponents, right._exponents, G.gens_orders()) ]\n+        exponents = [x + y for x, y in\n+                     zip(left._exponents, right._exponents)]\n         return G.element_class(G, exponents)\n \n     def __pow__(self, n):\n@@ -294,8 +302,7 @@\n         if n != m:\n             raise TypeError('argument n (= '+str(n)+') must be an integer.')\n         G = self.parent()\n-        exponents = [ (m*e) % order if order!=0 else m*e\n-                      for e,order in zip(self._exponents, G.gens_orders()) ]\n+        exponents = [m * e for e in self._exponents]\n         return G.element_class(G, exponents)\n \n     def __invert__(self):\n@@ -319,8 +326,7 @@\n             (-1, 4)\n         \"\"\"\n         G = self.parent()\n-        exponents = [(-e) % order if order != 0 else -e\n-                     for e, order in zip(self._exponents, G.gens_orders())]\n+        exponents = [-e for e in self._exponents]\n         return G.element_class(G, exponents)\n \n     def is_trivial(self):\n", "issue": "AbelianGroupElement does not always reduce exponent vectors when needed\nIn SageMath 9.8:\r\n```\r\nsage: A = AdditiveAbelianGroup([3])\r\nsage: A([6]) == A([0])\r\nTrue\r\nsage: A([6]).vector()\r\n(0)\r\nsage: M = AbelianGroup([3])\r\nsage: M([6]) == M([0])  # output should be True\r\nFalse\r\nsage: M([6]).exponents()\r\n(6,)\r\n```\r\nThe last output contradicts the documentation of the `exponents()` method, which says\r\n```\r\n   OUTPUT:\r\n\r\n   A tuple of integers for an abelian group element. The integer can\r\n   be arbitrary if the corresponding generator has infinite order. If\r\n   the generator is of finite order, the integer is in the range from\r\n   0 (inclusive) to the order (exclusive).\r\n```\r\nThe `exponents()` method should be fixed (so that it reduces the exponents modulo the respective orders) and `_richcmp_()` should use this method.  (Alternatively, `_richcmp_()` and possibly other methods could do the reduction modulo the orders, but this is probably be more fragile.)\n", "before_files": [{"content": "\"\"\"\nBase class for abelian group elements\n\nThis is the base class for both\n:mod:`~sage.groups.abelian_gps.abelian_group_element` and\n:mod:`~sage.groups.abelian_gps.dual_abelian_group_element`.\n\nAs always, elements are immutable once constructed.\n\"\"\"\n\n\n###########################################################################\n#  Copyright (C) 2006 William Stein <wstein@gmail.com>\n#  Copyright (C) 2006 David Joyner  <wdjoyner@gmail.com>\n#  Copyright (C) 2012 Volker Braun  <vbraun.name@gmail.com>\n#\n#  Distributed under the terms of the GNU General Public License (GPL)\n#                  https://www.gnu.org/licenses/\n###########################################################################\n\nfrom sage.structure.element import MultiplicativeGroupElement\nfrom sage.misc.cachefunc import cached_method\nfrom sage.arith.all import GCD, LCM\nfrom sage.rings.integer import Integer\nfrom sage.rings.integer_ring import ZZ\nfrom sage.rings.infinity import infinity\nfrom sage.structure.richcmp import richcmp\n\n\nclass AbelianGroupElementBase(MultiplicativeGroupElement):\n    \"\"\"\n    Base class for abelian group elements\n\n    The group element is defined by a tuple whose ``i``-th entry is an\n    integer in the range from 0 (inclusively) to ``G.gen(i).order()``\n    (exclusively) if the `i`-th generator is of finite order, and an\n    arbitrary integer if the `i`-th generator is of infinite order.\n\n    INPUT:\n\n    - ``exponents`` -- ``1`` or a list/tuple/iterable of integers. The\n      exponent vector (with respect to the parent generators) defining\n      the group element.\n\n    - ``parent`` -- Abelian group. The parent of the group element.\n\n    EXAMPLES::\n\n        sage: F = AbelianGroup(3,[7,8,9])\n        sage: Fd = F.dual_group(names=\"ABC\")\n        sage: A,B,C = Fd.gens()\n        sage: A*B^-1 in Fd\n        True\n    \"\"\"\n\n    def __init__(self, parent, exponents):\n        \"\"\"\n        Create an element.\n\n        EXAMPLES::\n\n            sage: F = AbelianGroup(3,[7,8,9])\n            sage: Fd = F.dual_group(names=\"ABC\")\n            sage: A,B,C = Fd.gens()\n            sage: A*B^-1 in Fd\n            True\n        \"\"\"\n        MultiplicativeGroupElement.__init__(self, parent)\n        n = parent.ngens()\n        if exponents == 1:\n            self._exponents = tuple( ZZ.zero() for i in range(n) )\n        else:\n            self._exponents = tuple( ZZ(e) for e in exponents )\n            if len(self._exponents) != n:\n                raise IndexError('argument length (= %s) must be %s.'%(len(exponents), n))\n\n    def __hash__(self):\n        r\"\"\"\n        TESTS::\n\n            sage: F = AbelianGroup(3,[7,8,9])\n            sage: hash(F.an_element()) # random\n            1024\n        \"\"\"\n        return hash(self.parent()) ^ hash(self._exponents)\n\n    def exponents(self):\n        \"\"\"\n        The exponents of the generators defining the group element.\n\n        OUTPUT:\n\n        A tuple of integers for an abelian group element. The integer\n        can be arbitrary if the corresponding generator has infinite\n        order. If the generator is of finite order, the integer is in\n        the range from 0 (inclusive) to the order (exclusive).\n\n        EXAMPLES::\n\n            sage: F.<a,b,c,f> = AbelianGroup([7,8,9,0])\n            sage: (a^3*b^2*c).exponents()\n            (3, 2, 1, 0)\n            sage: F([3, 2, 1, 0])\n            a^3*b^2*c\n            sage: (c^42).exponents()\n            (0, 0, 6, 0)\n            sage: (f^42).exponents()\n            (0, 0, 0, 42)\n        \"\"\"\n        return self._exponents\n\n    def _libgap_(self):\n        r\"\"\"\n        TESTS::\n\n            sage: F.<a,b,c> = AbelianGroup([7,8,9])\n            sage: libgap(a**2 * c) * libgap(b * c**2)\n            f1^2*f2*f6\n        \"\"\"\n        from sage.misc.misc_c import prod\n        from sage.libs.gap.libgap import libgap\n        G = libgap(self.parent())\n        return prod(g**i for g,i in zip(G.GeneratorsOfGroup(), self._exponents))\n\n    def list(self):\n        \"\"\"\n        Return a copy of the exponent vector.\n\n        Use :meth:`exponents` instead.\n\n        OUTPUT:\n\n        The underlying coordinates used to represent this element.  If\n        this is a word in an abelian group on `n` generators, then\n        this is a list of nonnegative integers of length `n`.\n\n        EXAMPLES::\n\n            sage: F = AbelianGroup(5,[2, 3, 5, 7, 8], names=\"abcde\")\n            sage: a,b,c,d,e = F.gens()\n            sage: Ad = F.dual_group(names=\"ABCDE\")\n            sage: A,B,C,D,E = Ad.gens()\n            sage: (A*B*C^2*D^20*E^65).exponents()\n            (1, 1, 2, 6, 1)\n            sage: X = A*B*C^2*D^2*E^-6\n            sage: X.exponents()\n            (1, 1, 2, 2, 2)\n        \"\"\"\n        # to be deprecated (really, return a list??). Use exponents() instead.\n        return list(self._exponents)\n\n    def _repr_(self):\n        \"\"\"\n        Return a string representation of ``self``.\n\n        OUTPUT:\n\n        String.\n\n        EXAMPLES::\n\n            sage: G = AbelianGroup([2])\n            sage: G.gen(0)._repr_()\n            'f'\n            sage: G.one()._repr_()\n            '1'\n        \"\"\"\n        s = \"\"\n        G = self.parent()\n        for v_i, x_i in zip(self.exponents(), G.variable_names()):\n            if v_i == 0:\n                continue\n            if len(s) > 0:\n                s += '*'\n            if v_i == 1:\n                s += str(x_i)\n            else:\n                s += str(x_i) + '^' + str(v_i)\n        if s:\n            return s\n        else:\n            return '1'\n\n    def _richcmp_(self, other, op):\n        \"\"\"\n        Compare ``self`` and ``other``.\n\n        The comparison is based on the exponents.\n\n        OUTPUT:\n\n        boolean\n\n        EXAMPLES::\n\n            sage: G.<a,b> = AbelianGroup([2,3])\n            sage: a > b\n            True\n\n            sage: Gd.<A,B> = G.dual_group()\n            sage: A > B\n            True\n        \"\"\"\n        return richcmp(self._exponents, other._exponents, op)\n\n    @cached_method\n    def order(self):\n        \"\"\"\n        Return the order of this element.\n\n        OUTPUT:\n\n        An integer or ``infinity``.\n\n        EXAMPLES::\n\n            sage: F = AbelianGroup(3,[7,8,9])\n            sage: Fd = F.dual_group()\n            sage: A,B,C = Fd.gens()\n            sage: (B*C).order()\n            72\n\n            sage: F = AbelianGroup(3,[7,8,9]); F\n            Multiplicative Abelian group isomorphic to C7 x C8 x C9\n            sage: F.gens()[2].order()\n            9\n            sage: a,b,c = F.gens()\n            sage: (b*c).order()\n            72\n            sage: G = AbelianGroup(3,[7,8,9])\n            sage: type((G.0 * G.1).order())==Integer\n            True\n        \"\"\"\n        M = self.parent()\n        order = M.gens_orders()\n        L = self.exponents()\n        N = LCM([order[i]/GCD(order[i],L[i]) for i in range(len(order)) if L[i]!=0])\n        if N == 0:\n            return infinity\n        else:\n            return ZZ(N)\n\n    multiplicative_order = order\n\n    def _div_(left, right):\n        \"\"\"\n        Divide ``left`` and ``right``\n\n        TESTS::\n\n            sage: G.<a,b> = AbelianGroup(2)\n            sage: a/b\n            a*b^-1\n            sage: a._div_(b)\n            a*b^-1\n        \"\"\"\n        G = left.parent()\n        assert G is right.parent()\n        exponents = [ (x-y)%order if order!=0 else x-y\n                      for x, y, order in\n                      zip(left._exponents, right._exponents, G.gens_orders()) ]\n        return G.element_class(G, exponents)\n\n    def _mul_(left, right):\n        \"\"\"\n        Multiply ``left`` and ``right``\n\n        TESTS::\n\n            sage: G.<a,b> = AbelianGroup(2)\n            sage: a*b\n            a*b\n            sage: a._mul_(b)\n            a*b\n        \"\"\"\n        G = left.parent()\n        assert G is right.parent()\n        exponents = [ (x+y)%order if order!=0 else x+y\n                      for x, y, order in\n                      zip(left._exponents, right._exponents, G.gens_orders()) ]\n        return G.element_class(G, exponents)\n\n    def __pow__(self, n):\n        \"\"\"\n        Exponentiate ``self``\n\n        TESTS::\n\n            sage: G.<a,b> = AbelianGroup(2)\n            sage: a^3\n            a^3\n        \"\"\"\n        m = Integer(n)\n        if n != m:\n            raise TypeError('argument n (= '+str(n)+') must be an integer.')\n        G = self.parent()\n        exponents = [ (m*e) % order if order!=0 else m*e\n                      for e,order in zip(self._exponents, G.gens_orders()) ]\n        return G.element_class(G, exponents)\n\n    def __invert__(self):\n        \"\"\"\n        Return the inverse element.\n\n        EXAMPLES::\n\n            sage: G.<a,b> = AbelianGroup([0,5])\n            sage: a.inverse()  # indirect doctest\n            a^-1\n            sage: a.__invert__()\n            a^-1\n            sage: a^-1\n            a^-1\n            sage: ~a\n            a^-1\n            sage: (a*b).exponents()\n            (1, 1)\n            sage: (a*b).inverse().exponents()\n            (-1, 4)\n        \"\"\"\n        G = self.parent()\n        exponents = [(-e) % order if order != 0 else -e\n                     for e, order in zip(self._exponents, G.gens_orders())]\n        return G.element_class(G, exponents)\n\n    def is_trivial(self):\n        \"\"\"\n        Test whether ``self`` is the trivial group element ``1``.\n\n        OUTPUT:\n\n        Boolean.\n\n        EXAMPLES::\n\n            sage: G.<a,b> = AbelianGroup([0,5])\n            sage: (a^5).is_trivial()\n            False\n            sage: (b^5).is_trivial()\n            True\n        \"\"\"\n        return all(e==0 for e in self._exponents)\n", "path": "src/sage/groups/abelian_gps/element_base.py"}], "after_files": [{"content": "\"\"\"\nBase class for abelian group elements\n\nThis is the base class for both\n:mod:`~sage.groups.abelian_gps.abelian_group_element` and\n:mod:`~sage.groups.abelian_gps.dual_abelian_group_element`.\n\nAs always, elements are immutable once constructed.\n\"\"\"\n\n\n###########################################################################\n#  Copyright (C) 2006 William Stein <wstein@gmail.com>\n#  Copyright (C) 2006 David Joyner  <wdjoyner@gmail.com>\n#  Copyright (C) 2012 Volker Braun  <vbraun.name@gmail.com>\n#\n#  Distributed under the terms of the GNU General Public License (GPL)\n#                  https://www.gnu.org/licenses/\n###########################################################################\n\nfrom sage.structure.element import MultiplicativeGroupElement\nfrom sage.misc.cachefunc import cached_method\nfrom sage.arith.all import GCD, LCM\nfrom sage.rings.integer import Integer\nfrom sage.rings.integer_ring import ZZ\nfrom sage.rings.infinity import infinity\nfrom sage.structure.richcmp import richcmp\n\n\nclass AbelianGroupElementBase(MultiplicativeGroupElement):\n    \"\"\"\n    Base class for abelian group elements\n\n    The group element is defined by a tuple whose ``i``-th entry is an\n    integer in the range from 0 (inclusively) to ``G.gen(i).order()``\n    (exclusively) if the `i`-th generator is of finite order, and an\n    arbitrary integer if the `i`-th generator is of infinite order.\n\n    INPUT:\n\n    - ``exponents`` -- ``1`` or a list/tuple/iterable of integers. The\n      exponent vector (with respect to the parent generators) defining\n      the group element.\n\n    - ``parent`` -- Abelian group. The parent of the group element.\n\n    EXAMPLES::\n\n        sage: F = AbelianGroup(3,[7,8,9])\n        sage: Fd = F.dual_group(names=\"ABC\")\n        sage: A,B,C = Fd.gens()\n        sage: A*B^-1 in Fd\n        True\n    \"\"\"\n\n    def __init__(self, parent, exponents):\n        \"\"\"\n        Create an element.\n\n        EXAMPLES::\n\n            sage: F = AbelianGroup(3,[7,8,9])\n            sage: Fd = F.dual_group(names=\"ABC\")\n            sage: A,B,C = Fd.gens()\n            sage: A*B^-1 in Fd\n            True\n\n        Check that :issue:`35216` is fixed::\n\n            sage: M = AbelianGroup([3])\n            sage: M([5]) == M([2])\n            True\n            sage: M([3]).is_trivial()\n            True\n        \"\"\"\n        MultiplicativeGroupElement.__init__(self, parent)\n        n = parent.ngens()\n        if exponents == 1:\n            self._exponents = tuple(ZZ.zero() for i in range(n))\n        else:\n            if len(exponents) != n:\n                raise IndexError('argument length (= %s) must be %s'\n                                 % (len(exponents), n))\n            self._exponents = tuple(ZZ(e % o if o else e) for e, o in\n                                    zip(exponents, parent.gens_orders()))\n\n    def __hash__(self):\n        r\"\"\"\n        TESTS::\n\n            sage: F = AbelianGroup(3,[7,8,9])\n            sage: hash(F.an_element()) # random\n            1024\n        \"\"\"\n        return hash(self.parent()) ^ hash(self._exponents)\n\n    def exponents(self):\n        \"\"\"\n        The exponents of the generators defining the group element.\n\n        OUTPUT:\n\n        A tuple of integers for an abelian group element. The integer\n        can be arbitrary if the corresponding generator has infinite\n        order. If the generator is of finite order, the integer is in\n        the range from 0 (inclusive) to the order (exclusive).\n\n        EXAMPLES::\n\n            sage: F.<a,b,c,f> = AbelianGroup([7,8,9,0])\n            sage: (a^3*b^2*c).exponents()\n            (3, 2, 1, 0)\n            sage: F([3, 2, 1, 0])\n            a^3*b^2*c\n            sage: (c^42).exponents()\n            (0, 0, 6, 0)\n            sage: (f^42).exponents()\n            (0, 0, 0, 42)\n        \"\"\"\n        return self._exponents\n\n    def _libgap_(self):\n        r\"\"\"\n        TESTS::\n\n            sage: F.<a,b,c> = AbelianGroup([7,8,9])\n            sage: libgap(a**2 * c) * libgap(b * c**2)\n            f1^2*f2*f6\n        \"\"\"\n        from sage.misc.misc_c import prod\n        from sage.libs.gap.libgap import libgap\n        G = libgap(self.parent())\n        return prod(g**i for g,i in zip(G.GeneratorsOfGroup(), self._exponents))\n\n    def list(self):\n        \"\"\"\n        Return a copy of the exponent vector.\n\n        Use :meth:`exponents` instead.\n\n        OUTPUT:\n\n        The underlying coordinates used to represent this element.  If\n        this is a word in an abelian group on `n` generators, then\n        this is a list of nonnegative integers of length `n`.\n\n        EXAMPLES::\n\n            sage: F = AbelianGroup(5,[2, 3, 5, 7, 8], names=\"abcde\")\n            sage: a,b,c,d,e = F.gens()\n            sage: Ad = F.dual_group(names=\"ABCDE\")\n            sage: A,B,C,D,E = Ad.gens()\n            sage: (A*B*C^2*D^20*E^65).exponents()\n            (1, 1, 2, 6, 1)\n            sage: X = A*B*C^2*D^2*E^-6\n            sage: X.exponents()\n            (1, 1, 2, 2, 2)\n        \"\"\"\n        # to be deprecated (really, return a list??). Use exponents() instead.\n        return list(self._exponents)\n\n    def _repr_(self):\n        \"\"\"\n        Return a string representation of ``self``.\n\n        OUTPUT:\n\n        String.\n\n        EXAMPLES::\n\n            sage: G = AbelianGroup([2])\n            sage: G.gen(0)._repr_()\n            'f'\n            sage: G.one()._repr_()\n            '1'\n        \"\"\"\n        s = \"\"\n        G = self.parent()\n        for v_i, x_i in zip(self.exponents(), G.variable_names()):\n            if v_i == 0:\n                continue\n            if len(s) > 0:\n                s += '*'\n            if v_i == 1:\n                s += str(x_i)\n            else:\n                s += str(x_i) + '^' + str(v_i)\n        if s:\n            return s\n        else:\n            return '1'\n\n    def _richcmp_(self, other, op):\n        \"\"\"\n        Compare ``self`` and ``other``.\n\n        The comparison is based on the exponents.\n\n        OUTPUT:\n\n        boolean\n\n        EXAMPLES::\n\n            sage: G.<a,b> = AbelianGroup([2,3])\n            sage: a > b\n            True\n\n            sage: Gd.<A,B> = G.dual_group()\n            sage: A > B\n            True\n        \"\"\"\n        return richcmp(self._exponents, other._exponents, op)\n\n    @cached_method\n    def order(self):\n        \"\"\"\n        Return the order of this element.\n\n        OUTPUT:\n\n        An integer or ``infinity``.\n\n        EXAMPLES::\n\n            sage: F = AbelianGroup(3,[7,8,9])\n            sage: Fd = F.dual_group()\n            sage: A,B,C = Fd.gens()\n            sage: (B*C).order()\n            72\n\n            sage: F = AbelianGroup(3,[7,8,9]); F\n            Multiplicative Abelian group isomorphic to C7 x C8 x C9\n            sage: F.gens()[2].order()\n            9\n            sage: a,b,c = F.gens()\n            sage: (b*c).order()\n            72\n            sage: G = AbelianGroup(3,[7,8,9])\n            sage: type((G.0 * G.1).order())==Integer\n            True\n        \"\"\"\n        M = self.parent()\n        order = M.gens_orders()\n        L = self.exponents()\n        N = LCM([order[i]/GCD(order[i],L[i]) for i in range(len(order)) if L[i]!=0])\n        if N == 0:\n            return infinity\n        else:\n            return ZZ(N)\n\n    multiplicative_order = order\n\n    def _div_(left, right):\n        \"\"\"\n        Divide ``left`` and ``right``\n\n        TESTS::\n\n            sage: G.<a,b> = AbelianGroup(2)\n            sage: a/b\n            a*b^-1\n            sage: a._div_(b)\n            a*b^-1\n        \"\"\"\n        G = left.parent()\n        assert G is right.parent()\n        exponents = [x - y for x, y in\n                     zip(left._exponents, right._exponents)]\n        return G.element_class(G, exponents)\n\n    def _mul_(left, right):\n        \"\"\"\n        Multiply ``left`` and ``right``\n\n        TESTS::\n\n            sage: G.<a,b> = AbelianGroup(2)\n            sage: a*b\n            a*b\n            sage: a._mul_(b)\n            a*b\n        \"\"\"\n        G = left.parent()\n        assert G is right.parent()\n        exponents = [x + y for x, y in\n                     zip(left._exponents, right._exponents)]\n        return G.element_class(G, exponents)\n\n    def __pow__(self, n):\n        \"\"\"\n        Exponentiate ``self``\n\n        TESTS::\n\n            sage: G.<a,b> = AbelianGroup(2)\n            sage: a^3\n            a^3\n        \"\"\"\n        m = Integer(n)\n        if n != m:\n            raise TypeError('argument n (= '+str(n)+') must be an integer.')\n        G = self.parent()\n        exponents = [m * e for e in self._exponents]\n        return G.element_class(G, exponents)\n\n    def __invert__(self):\n        \"\"\"\n        Return the inverse element.\n\n        EXAMPLES::\n\n            sage: G.<a,b> = AbelianGroup([0,5])\n            sage: a.inverse()  # indirect doctest\n            a^-1\n            sage: a.__invert__()\n            a^-1\n            sage: a^-1\n            a^-1\n            sage: ~a\n            a^-1\n            sage: (a*b).exponents()\n            (1, 1)\n            sage: (a*b).inverse().exponents()\n            (-1, 4)\n        \"\"\"\n        G = self.parent()\n        exponents = [-e for e in self._exponents]\n        return G.element_class(G, exponents)\n\n    def is_trivial(self):\n        \"\"\"\n        Test whether ``self`` is the trivial group element ``1``.\n\n        OUTPUT:\n\n        Boolean.\n\n        EXAMPLES::\n\n            sage: G.<a,b> = AbelianGroup([0,5])\n            sage: (a^5).is_trivial()\n            False\n            sage: (b^5).is_trivial()\n            True\n        \"\"\"\n        return all(e==0 for e in self._exponents)\n", "path": "src/sage/groups/abelian_gps/element_base.py"}]}