Let $G$ be a group. If there exists a positive integer $m$ such that $a^m=1$ for all $a\in G$, then is $G$ finite?
6189
Examples of infinite groups such that all their respective elements are of finite order.
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
26148
A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian?
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
2104
Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
152071
Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
22783
Is the set $Z_n = \{ 0, 1, . . . , n-1 \}$ for $n ≥ 1$ is a group under addition modulo n?
146312
Is the set of roots with respect to their values and multiplicies unique for each $g(x)$ in $a_n x^n + g(x)$?
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
7079
ord $(b)|\max\{\text{ord}(g)|g\in G\}$ for all $b\in G\,$ a finite abelian group
43750
Subgroups of finite solvable groups. Solvable?
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$