Let $G:=(\Bbb Z,*)$ where $a*b:=a+(-1)^ab$ for any $a,b\in\Bbb Z$. Show that $G$ is a group.
93031
How to show that $\mathbb{Z}$ with the operation $m*n=m+(-1)^m n$ is associative?
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
29717
Not sure how to go about this group theory proof
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
117701
Let $G$ be a group and $X$ set. $a,b \in X$ are in the same orbit, so show that $stab(a) \cong stab (b)$
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
32735
If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
120642
Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian