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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Is this inverse image nontrivial?

I would like to prove that if a non-comutative group $G$ has a subgroup $H$ whose index is equal to $3$ or $4$, then $G$ is not simple. In order to do, I took a group action $$\phi : G \times G/H \ni (...
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2 votes
0 answers
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Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
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2 votes
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Show that $\langle a\rangle$ is a normal subgroup of $\langle a,b\rangle$ where $a$ and $b$ are permutations

I'm studying for an exam and in a previous year we had the question: Let $a=(12345678)$, and $b=(26)(48)$ and $G=\langle a,b \rangle$. Show that $\langle a \rangle$ is a normal subgroup of $G$ and by ...
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1 answer
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On nonsplit noncentral extension of finite simple groups

Let $G$ be a finite group and $N$ be its minimal normal subgroup such that $G/N$ is a finite simple group. It is well know that if $G=G'$ and $N=Z(G)$, then $N=\Phi(G)$. My Question is: Is it true ...
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1 vote
1 answer
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What are the conditions for cosets of a certain group to form a group themselves?

I'm currently watching Scoratica's YouTube series on Group Theory. On one of the videos, the following argument is made ($N$ is a normal subgroup of $G$): Condition for the cosets to act like a group ...
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-1 votes
0 answers
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Bourbaki Algebraic Structures Question 4.8

Second part unsolved. Let $\mathrm A$ and $\mathrm A'$ be two groups and $\mathrm G$ a subgroup of $\mathrm{A\times A'}$. We write: $\mathrm{N = G \cap (A \times \{e\}), H = pr_1(G)}$ $\mathrm{N' = G \...
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0 votes
1 answer
67 views

Infinite group with finitely many normal subgroups

In theorem 2 of this, a finite number of non normal group is classified. I want to know the opposite of this question. Is there any classification of groups with finitely many normal subgroups?
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When should Sylow subgroups intersect and when they should not?

Here is the question I am trying to understand its solution: Prove that a group of order $11 \times 2^{10}$ has a normal subgroup. And here is a solution I found to the part of excluding the case $n_2 ...
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-3 votes
1 answer
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Excluding the case $n_2 = 11$ and $n_{11} = 1024.$ [duplicate]

Here is the question I am trying to solve: Prove that a group of order $11 \times 2^{10}$ has a normal subgroup. And here is a solution I found to the part of excluding the case $n_2 = 11$ and $n_{11} ...
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2 votes
1 answer
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Is every subgroup in a solvable group's composition series, also a normal subgroup?

I am trying to understand this proof: https://math.stackexchange.com/a/1889305/74378 To summarize what is going on there, $G$ is a solvable group with composition series $$ 1\triangleleft G_1 \...
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2 votes
1 answer
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Prove that a group of order $11 \times 2^{10}$ has a normal subgroup.

I am trying to solve this question: Prove that a group of order $11 \times 2^{10}$ has a nontrivial proper normal subgroup. My trial By Sylow theorems I know that $n_2 \in \{1,11\}$ and $n_{11} \in \{...
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0 votes
1 answer
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Is every normal subgroup of some group also a cyclic subgroup? [closed]

I know the converse doesn't hold, but I am unable to find a counterexample. Yet I don't know how normal subgroups bring rise to a generator.
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Non-Abelian group without subgroup in this particular form. [duplicate]

So for my math class (Introduction to Groups and Rings) and I have encountered the following question: $\hspace{15px}\|$Find an example of an non-abelian group $G$ such that $H=\{g\in G : |g|<\...
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0 votes
1 answer
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 ...
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2 votes
0 answers
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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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4 votes
1 answer
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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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1 vote
2 answers
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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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2 votes
1 answer
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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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0 votes
0 answers
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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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-1 votes
0 answers
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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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-1 votes
1 answer
66 views

Show that the function is homomorphism with kernel φ

If $H_1,\dots, H_s$ are normal subgroups of $G$, show that the function: $$\varphi: G\to G/H_1 \times\dots\times G/H_s$$ given by: $$\varphi(g) = (gH_1,\dots, gH_s)$$ is a homomorphism with kernel: $$\...
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1 vote
0 answers
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If a subgroup is closed under conjugation by an element then is it also closed under conjugation by its inverse? [duplicate]

This isn't my original problem, but I have proven that the problem I'm looking at can be reduced to this one. If a subgroup is closed under conjugation by an element then is it also closed under ...
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1 vote
0 answers
30 views

Show that the intersection of these two subgroups has prime index

Let the subgroup $N$ be normal in the finite group $G$ with index a prime $p$. Let $H$ be a subgroup of $G$ which is not contained in $N$. I would like to show that $H \cap N$ is normal in $H$ with ...
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0 votes
1 answer
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Generalization of Semidirect Product (Finite Groups with non-trivial minimal subgroup)

Suppose any two non-trivial (non-singleton) subgroups of a group $G$ have a non-trivial intersection. $|G|$ is necessarily a prime power, because by Cauchy's theorem, for any prime $q$ dividing $|G|$, ...
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-6 votes
1 answer
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Let $G$ be a group and $H\le G$. If there exists a homomorphism $f:G\to H$ such that $f(h)=h$ for all $h\in H$, then is $H$ normal in $G$? [closed]

Let $G$ be a group and $H$ be a subgroup of $G$. If there exists a homomorphism $f:G\to H$ such that $f(h)=h$ for all $h\in H$, then is $H$ normal?
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3 votes
1 answer
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Extension in semidirect case

Suppose $G=NQ$, where $N$ is normal in $G$ and $(|N|,|Q|)=1.$ In other words $G$ is a semidirect product of $N$ and $Q.$ Can we say every irreducible character of $N$ is extendable to $G?$ Suppose $\...
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0 votes
0 answers
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How to show that if $G/N$ and $N$ soluble then so is $G$ [duplicate]

I am trying to show that if $N$ is a normal subgroup of $G$ that is soluble and $G/N$ is soluble then so is $G$. I am wondering whether it has something to do with the isomorphism theorem for groups. ...
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0 votes
0 answers
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Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. If $G = Z(G)M$, then $M$ is normal in $G$

I need to prove that if $G$ is a group, $M$ is a maximal subgroup of $G$ and $Z(G) \nsubseteq M$,then $M \unlhd G$. Is true that $G = Z(G)M$, right? Is this enough?
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1 vote
1 answer
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Prove that if $H_1\lhd G, H_2\lhd G, H_1H_2=G, H_1\cap H_2=\{1\}$, then $G/H_1\simeq H_2$.

Problem: Suppose $G$ is a group, and $H_1, H_2$ are subgroups of $G$. Prove that if $H_1, H_2$ are normal subgroups of $G$, $H_1H_2=\{xy: x\in H_1, y\in H_2\}=G$, and $H_1\cap H_2=\{1\}$ where $1$ is ...
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0 votes
1 answer
62 views

Intersection of all Sylow $p$-subgroups

I wanted to prove that the intersection of all Sylow $p$-subgroups of a finite group G is a normal subgroup of $G$. Can someone enlighten me how is this implication possible: If an automorphism $\...
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2 votes
1 answer
83 views

Is there a convenient way to find $[G,G]$?

I am concerned with finding the commutator subgroup $[G,G]$ of a finite group $G$. Recall that the commutator subgroup $[G,G]$ of a finite group $G$ is given by $[G,G] = \langle\{g^{-1}h^{-1}gh: g,h \...
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-6 votes
1 answer
68 views

Let $G$ act on $X$. Show $N= \{ a \in G \mid ax = x \ \mbox{for all} \ \ x \in X \}.$ Show that $N\unlhd G$. [closed]

Suppose $G$ acts on $X$. Let $$N = \{ a \in G \mid ax = x \ \mbox{for all} \ \ x \in X \}.$$ Show that $N$ is a normal subgroup of $G.$ Not sure where to go with this one
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0 votes
0 answers
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A minimal prime order of a group. [duplicate]

I have a group $G$ such that there's a minimal prime number $p$ that divides $|G|$, and I am given a subgroup $H$ in $G$, such that $[G:H]=p$. I would like to prove that $H$ is normal in $G$. I know ...
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0 votes
1 answer
18 views

A weak condition for normality of a subgroup.

Is it true that $N$ is a normal subgroup of $G$ iff $\forall g\in G \exists x\in G : gN=Nx$? I don't think so, but I can't think of a counterexample obviously if $N$ is normal then the above condition ...
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1 vote
1 answer
52 views

Show $\forall h\in H,\forall g\in G, ghg^{-1}=h$ is not definition of normal subgroup.

Let $G$ be a group and $H$ be subgroup of $G$. $H$ is normal subgroup if only if $$\forall h\in H,\forall g\in G, ghg^{-1}\in H.\tag{1}$$ I want to know a lot of examples $H$ which does not satisfy $$\...
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3 votes
0 answers
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$p$-subgroup is a proper subset of its normalizer

I'm trying to understand the following proof from Kurzweil-Stellmacher: How does $PgP=Pg$ imply $gPg^{-1}=P$? Transcription of image: 3.1.10 Let $P$ be a $p$-subgroup of $G$ and $p$ be a divisor of $...
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0 answers
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Questions clarification needed for showing that $H$ is an abelian group of odd order.

For the following exercise, I have some question about the notation and hint: Let $(G, *)$ be a finite group of even order. Suppose that half of the elements of $G$ are of order 2 and the rest of the ...
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  • 1,529
0 votes
2 answers
59 views

Quotient of a minimal normal subgroup is maximal subgroup provided complement exists? [closed]

Let $G$ be group and $N$ be a minimal normal subgroup of $G$ such that its complement exist in $G$ (say $K$). Is it true that $K$ is a maximal subgroup of $G$ ? Refer this for definition of complement....
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  • 335
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0 answers
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Find specific example for where the case works (topic: isomorphism and normal subgroups)

Consider groups $G_i, K_i, F_i$ such that $K_i \triangleleft G_i$ and $G_i / K_i\cong F_i$ , for $i = 1, 2$. In each case, find an example with the given properties or prove that no such example ...
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  • 409
1 vote
1 answer
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$G$ has an abelian subgroup $K$ with $|K|\geq p^3$.

Let $G$ be a group of order $p^4$ for some prime number $p$. Suppose $G$ has a normal subgroup $H$ of order $p^2$. Then $G$ has an abelian subgroup $K$ with $|K|\geq p^3$. First, by acting $G$ onto $...
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0 votes
0 answers
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 $N$ ...
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  • 1,529
0 votes
2 answers
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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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  • 1,529
3 votes
1 answer
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:(\...
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  • 4,119
3 votes
1 answer
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On isomorphic normal subgroups of a group

Well this is quite stupid a question because the intuition is false. See comments and answers below. Original question: Consider $H_1,H_2\le G$, $H_1 \cong H_2$. It seems trivial that if $H_1 \unlhd ...
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1 answer
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Action on Quotient group

Let $G =\langle A \rangle \leq S_n$ be a permutation group. The natural action of $G$ on $\Omega$ , where $|\Omega|=n$ is defined as follows: $f:G \times \Omega \rightarrow \Omega$ such that $f(\sigma,...
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0 votes
0 answers
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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 ...
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  • 49
2 votes
1 answer
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 ...
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  • 171
0 votes
1 answer
79 views

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 \...
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