Verifying $G*H \cong G' * H' \implies |G| = |G'|$ or $|G| = |H'|$ (All Groups Cyclic)
98155
Free products of cyclic groups
27436
Does $G\cong G/H$ imply that $H$ is trivial?
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
98542
Prove that $H$ is a subgroup of an abelian group $G$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
3462
Proving a set of automorphisms of a group is a group under composition
58766
free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$
35188
Counterexample: $G \times K \cong H \times K \implies G \cong H$
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
48585
Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
52095
Prove either $G=ST$ or |$G|\geq|S|+|T|$
124150
Faithful representation implies group is cyclic
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
129485
A subgroup of a cyclic group is cyclic - Understanding Proof
135422
$\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A') \Rightarrow A \cong A'$, proof verification