Let $G$ be a finite group and $Z(G):=\{x\in G : gx=xg \ \ \forall g\in G\}.$ Prove that $[G:Z(G)]$ is not a prime number.
7528
Index of the center of a group in the group is not a prime number
11003
Proving that $G/N$ is an abelian group
71420
Prove that a group is abelian.
106123
Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?
60804
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
79995
G is group of order pq, pq are primes
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
93029
Prove that when $d$ is in $\mathbb{Q} \setminus\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}]$ is not a finitely generated abelian group.
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
102622
K is a subgroup of G. Show that K is not a normal subgroup
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
59128
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
2254
Finite group with isomorphic normal subgroups and non-isomorphic quotients?
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
25050
To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.