# Questions tagged [normal-subgroups]

For questions concerning normal subgroups of groups. Consider using with the (group-theory) and/or the (abstract-algebra) tags too.

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### $H \lhd G$, $\pi:G \to H$ is a group homomorphism with $\pi(h)=h$ show that $G \cong H × G/H$

Let $H$ be a normal subgroup of a group $G$ such that there is a group homomorphism $\pi:G \to H$ with $\pi(h)=h$ for all $h \in H$. Prove that $G$ is isomorphic to $H × G/H$ Don't know how to solve ...
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### Proof verification: if $|G|/|H| = 2$ then $H$ is normal.

I'm trying to prove that if $H$ is a subgroup of $G$ with $|G|/|H| = 2$, then $H$ is normal. The problem statement doesn't specify whether these are left or right cosets of $H$ in $G$, but I don't ...
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### Word problem in residually finite groups - enumerating normal subgroups

I am trying to prove that the word problem is solvable for residually finite, finitely presented groups. It is known that one runs two algorithms in parallel. One algorithm stops when the word $w$ is ...
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### Example of $H$, $N$ two normal subgroups of a group G such that $G/N \cong G/H$ but $N \ncong H$ [duplicate]

I have tried to prove that if $H$, $N$ are two normal subgroups of a group G such that $G/N \cong G/H$, then $N \cong H$. I think that it is not possible, but I can't find a counterexample. What is an ...
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### Isomorphism between group mod inclusion and cartesian product mod normal subgroup

Given $G_1, G_2$ Groups and $N$ a normal subgroup of $G_1\times G_2$, I have already proved that the projections $p_i: G_1\times G_2\to G_i$ and $i_1:g_1\mapsto (g_1,1), \; i_2:g_2\mapsto (1, g_2)$ ...
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### Let $G$ be a group with $K\unlhd G$ and Abelian $N\unlhd G$. Is $G/(NK)$ an Abelian group?

Let $G$ be a group such that $N$ is an Abelian normal subgroup of $G$ and $K$ is just a normal subgroup of $G$, is $G/(NK)$ an Abelian group? Also we know that $NK$ is normal in $G$. I am not really ...
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### Exponent of the commutator factor group of a p-group

This is a follow up of a question I asked recently here Center and abelianization of a finite p-group. Many thanks to Arturo for a very detailed answer by the way. Let $P$ be a normal subgroup of a ...
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### Normal subgroups chains and symmetries of geometric shapes

There is a beautiful parallel between the normal subgroups chain of symmetric groups and the symmetries of 2D/3D shapes. Here's the tables for each symmetric group. S2: rotation of 2 vertices of an &...
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### What does this proposition state regarding normal subgroups?

If $K\triangleleft G$ so the subgroups from $G/K$ are quotient groups $H/K$ with $K<H<G$ and $H/K \triangleleft G/K$ is equivalent to $H \triangleleft G$. I didn't understand "are quotient ...
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### Question about the hypothesis for the question: if $N=\cap K$ then prove that N is a normal subgroup of $G$

For the following question: Let $G$ be a group that contains at least one subgroup of order $n.$ Let $N=\cap K,$ where the intersection is taken over all subgroups $K$ of order $n$. Prove that $N$ ...
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### Question about isomorphism between the subgroups $N$ and $a^{-1}Na$

In showing isomorphism between the subgroups $N$ and $a^{-1}Na$ of a group $G$, one usually define a function $f:N\rightarrow a^{-1}Na$ by $f(n)=a^{-1}na$ for all $a\in G$ and some $n\in N$. This ...
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### Pointwise stabiliser of a union of orbits is a normal subgroup

Let a group $G$ act on a set $S$. Define the set orbit and pointwise stabiliser for all $A\subseteq S$, \begin{align*} G\cdot A&:=\{g\cdot a:g\in G,a\in A\}\\ \textrm{Stabp}_G(A)&:=\{g\in G:(\...
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### Let $K \lhd G$. Then $|G/K| = p$ prime iff $K\le H \le G \Rightarrow H=K$ or $H = G$ [duplicate]

Let $K \lhd G$. Prove are equivalent: a) $K\le H \le G \Rightarrow H=K$ or $H = G$ b) $|G/K| = p$ with $p$ prime This is what I know so far about the problem $K \lhd G$ (stands for '$K$ is normal ...
Let $G$ be a group and $H, N \subseteq G$ be subsets of the group. Let $\overline{A}$ denote the normal closure of any subset $A\subseteq G'$ in some group $G'$. Let $\pi: G \to G/\overline{N}$ denote ...
### Let $H\le G, g\in G$ with order $n$, and $gH\in G / H$ with order $d$. Show $d$ divides $n$.
Let $H$ be a normal subgroup of $G$. Now, $g\in G$ has order $n$ and $gH\in G / H$ has order $d$. Show that $d$ divides $n$. So, $H\le G$, $|g|=n$, and $|gH|=d$ If $d\mid n$, then $dt=n$, some \$t\in \...