If G is a group and $|Aut(G)|=1$then prove G’=(e)
12455
$|G|>2$ implies $G$ has non trivial automorphism
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
96910
Prove the following alternative group definition.
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
151621
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
43059
Show that $g^k=e$ if and only if the order of $g$ is finite and divides $k$.
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
151559
Please help: How to show that Aut(G) is a subgroup of Sym(G)
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
114522
Prove that identity is unique in a group
105471
How to prove that if each element of group is inverse to itself then group commutative?
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?