Let G be a Group and $a,b \in G$. If $a$ and $b$ commute and $|a|$ and $|b|$ are finite. What can be possible values for $|ab|$?
37061
The order of $ab$ when $a,b$ commute
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
145254
$G$ be a non-nilpotent and supersoluble finite group.
145159
Finite group is abelian if the representatives of its conjugacy classes commute
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
51352
Subgroups of a finite abelian group
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.
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
13355
Product of elements of a finite abelian group
148569
Embeddings $A → B → A$, but $A \not\cong B$?
39469
If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
117538
What is the interpretation of $a \equiv b$ mod $H$ in group theory?
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
65613
$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.