Questions tagged [normal-subgroups]

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

2,228 questions
Filter by
Sorted by
Tagged with
45 views

• 4,300
148 views

106 views

$H \lhd G$, $\pi:G \to H$ 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 ...
38 views

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 ...
• 307
48 views

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 ...
• 41
1 vote
49 views

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 ...
• 95
29 views

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 $i_1:g_1\mapsto (g_1,1), \; i_2:g_2\mapsto (1, g_2)$ ...
• 257
50 views

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 ...
40 views

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 ...
• 21
16 views

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 &...
40 views

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 ...
66 views

• 398
33 views

33 views

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 $N$ ...
• 1,529
47 views

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 ...
• 1,529
57 views

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:(\...
• 4,119
93 views

• 335
58 views

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 ...
• 49
36 views

Quotient group by normal closure of union

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 ...
• 171
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 \...