prove that if $k|m$, $\mathbb{Z}_m$ has a subgroup of order $k$
97833
Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$
109053
Show that a finite cyclic group of order $n$ has exactly one subgroup of order $d$ dividing $n$ and these are all the subgroups it has
51747
Showing that the intersection of all subgroups of order $n$ is normal
121423
Prove that every element of a finite group has an order
133770
Subgroups of finite abelian groups.
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
81543
Finite abelian group contains an element with order equal to the lcm of the orders of its elements
53409
Show group of order $4n + 2$ has a subgroup of index 2.
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
108830
Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
137739
Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
40513
A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
64408
Subgroup of $\mathbb{Q}$ with finite index
51352
Subgroups of a finite abelian group