Prove that $G$ is abelian and has odd order
44361
An automorphism that has no fixed points except for the identity and is its own inverse implies commutativity
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
86823
proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
126893
A group of order 2p (p prime) and other conditions - prove abelian.
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
138862
Product of all elements in an odd finite abelian group is 1
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
1577
Order of nontrivial elements is 2 implies Abelian group
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
114590
Prove that any finite group $G$ of even order contains an element of order $2.$
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
62455
Prove Equivalence Relation in G
81543
Finite abelian group contains an element with order equal to the lcm of the orders of its elements
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
152630
showing that a group of order 45 is abelian
113936
Showing that a finite abelian group has a subgroup of order m for each divisor m of n