Let $(G,\cdot)$ be a group and $a\in G$. Define $x\ast y:=x\cdot a \cdot y$. Show $(G,\ast)$ is a group.
24181
Showing that $G$ is a group under an alternative operation.
71420
Prove that a group is abelian.
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
151559
Please help: How to show that Aut(G) is a subgroup of Sym(G)
101596
Determine group G as cyclic
70034
Showing that this set of functions is a group.
114221
A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
92430
Show that the group is cyclic.
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
103216
The set $\{g^2 | g \in G\}$ in a group $G$
152604
For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
126672
Product of principal ideals: $(a)\cdot (b) = (a b)$
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
122330
Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.
32735
If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.