Every finite group of order more than two has a nontrivial automorphism
12455
$|G|>2$ implies $G$ has non trivial automorphism
1577
Order of nontrivial elements is 2 implies Abelian group
150760
Are there any distinct finite simple groups with the same order?
88718
A group is generated by two elements of order $2$ is infinite and non-abelian
83710
Order of a homomorphism of groups
47478
Show that any cyclic group of even order has exactly one element of order $2$
18182
An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
36147
A finite group is nilpotent iff two elements with relatively prime order commute
101735
A finite $p$-group cannot be simple unless it has order $p$
15341
Non-abelian finite group in which more than half of the elements have order $2$
133664
When the automorphism group is trivial......
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
96878
A finite group with the property that all of its proper subgroups are abelian
3462
Proving a set of automorphisms of a group is a group under composition
42582
Infinite group with only two conjugacy classes
72991
Is every group the automorphism group of a group?
102429
There are 2 groups of order 6 (up to isomorphism)