Let $G$ be a group. Prove that if $|G|=p^n$ with $p$ prime and $n \in \mathbb{N}$ then $C(G) \neq {1}$
69820
A nontrivial p-group has nontrivial center
72424
Suppose 1 ≤ k ≤ n − 1 and gcd(k, n) = 1. Prove that gcd(n − k, n) = 1
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
59128
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
137186
Let G be $\mathbb Z_p\times\dots\times \mathbb Z_p$ . Find A(G).
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
9110
Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
137739
Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
126893
A group of order 2p (p prime) and other conditions - prove abelian.
114221
A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?