Let $G$ be a finite group with cardinality not divisible by $3$. How can one show that for every $g \in G$ there is an $h \in G$ such that $g=h^3$?
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
122330
Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
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 $ab\in H$.
156319
Show that a group with $|G| = 33$ contains an element of order $3$
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
64193
Cardinality of Infinite Symmetric group
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
135701
An example of a group such that $G \cong G \times G$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
128300
Prove that a group with exactly two proper nontrivial subgroups is isomorphic to $\mathbb{Z}_{pq}$ or $\mathbb{Z}_{p^3}$.
129120
Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them.
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.