# Questions tagged [quotient-group]

This tag is for questions relating to "Quotient Group".

830 questions
Filter by
Sorted by
Tagged with
44 views

48 views

### Finding all the possible orders of elements in a quotient group

Let $A$ be an arbitrary Abelian group and let $H:=\{a\in A\mid\exists b\in A\mid a=b^3\}.$ Prove $H$ is a normal subgroup of $A$. Determine all the possible orders of elements in the group $A/H.$ My ...
1 vote
52 views

### $G/H \cong \mathbb{Z}$ implies existence of $K \leq G$ s.t. $K \cap H = 1$ and $HK = G$

Let $H \lhd G$ and $G/H \cong \mathbb{Z}$. The claim is that there is a subgroup $K \leq G$ such that $K \cap H = 1$ and $HK = G$. Is the following correct? Take $a$ to be any element in a generator ...
36 views

### Stallings' Theorem

I am trying to understand Stallings' Theorem for lower central series. Here is the statement: Say we have groups $A, B$ with lower central series $A=A_1, A_2, ...$ and $B=B_1, B_2, ...$ respectively. ...
99 views

### Quotient group by two isomorphic groups

In the processes of studying some questions I suddenly realized something basic but weird. We have $\mathbb{Z}\cong 2\mathbb{Z}$, but $\mathbb{Z}/\mathbb{Z}\ncong \mathbb{Z}/2\mathbb{Z}$. It looks ...
133 views

### Composition series with non isomorphic quotients

Question Let $1\lhd G_1 \lhd \ldots \lhd G_n=G$ be a composition series of the group $G$. If for every $i\not= j$ the quotients $G_{i+1}/G_i$ and $G_{j+1}/G_j$ are non isomorphic, then show that every ...
1 vote
49 views

1 vote
53 views

### When do $H_1 \cong H_2$ and $K_1 \cong K_2$ imply $F(H_1,K_1) \cong F(H_2,K_2)$?

Suppose $H_1$ and $K_1$ are subgroups of $G_1$ and $H_2$ and $K_2$ are subgroups of $G_2$ such that $H_1 \cong H_2$ and $K_1 \cong K_2$. When can we say $H_1 \cap K_1 \cong H_2 \cap K_2$? If this ...
### How to I take the quotient $GL^+(2,\mathbb{R})/SO(2,\mathbb{R})$
I am using the following representation of the $GL^+(2,\mathbb{R})$ group.  \exp( a+x \sigma_1 +y \sigma_2 + b \sigma_1\sigma_2) = \exp( \begin{bmatrix}a+x & -b +y \\ b+y & a-x\end{bmatrix}...