List questions from quotient-group | MathQa

Let $ G $ be a group and $ N \lhd G $. Why is $G $ supersoluble if $ N $ is cyclic and $ G/N $ is supersoluble?

Published on 13 Jan 2025 - 2:34

Show a homomorphism between groups induces a homomorphish between their respective quotient groups

Published on 13 Jan 2025 - 0:06

Is there an equivalence between representations of $N$ and $G/N$, and a representation of $G$?

Published on 13 Jan 2025 - 0:03

Let $L$ be a subgroup of $\mathbb{Z}^3$ of index $16$. What are the possibilities for $\mathbb{Z}^3 /L$?

Published on 13 Jan 2025 - 0:04

Let $G=\langle X\mid R\rangle$ and $H=\langle X\mid S\rangle$. If $R\subseteq S$, the $H$ is isomorphic to a quotient of $G$.

Published on 13 Jan 2025 - 0:08

If $G/Z(G)$ is cyclic, why is $G$ only abelian and not also cyclic?

Published on 12 Jan 2025 - 23:52

Question about preimage of conjugate class of quotient group

Published on 12 Jan 2025 - 23:52

Quotient of free groups $\langle a_1,\ldots, a_n\rangle /\langle a_2-a_1,a_3-a_2,\ldots,a_{n}-a_{n-1},a_1-a_n\rangle$

Published on 13 Jan 2025 - 0:07

Let $G$ be a group, and $M$, $N$ be normal subgroups such that $M \leq N$. If $G/N \cong G/M$ then need it be true that $N = M$?

Published on 13 Jan 2025 - 0:07

Suppose $F$ is a finitely generated free group, with $M,N \trianglelefteq F$ such that $M \leq N$. If $F/N \cong F/M$ then need we have $N=M$?

Published on 13 Jan 2025 - 0:08