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Questions tagged [normal-subgroups]

For questions concerning normal subgroups of groups.

3
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2answers
65 views

multiplying two normal subgroups is still normal?

Let $G_i \triangleleft G_{i+1}$ both subgroups of $G$. Let $N$ be a normal group. Does $G_iN \triangleleft G_{i+1}N$? Does $(G_iN/N) \triangleleft (G_{i+1}N/N)$? I know that $q:G\longrightarrow G/...
0
votes
3answers
82 views

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $ G\cong N \times G/N. $ [duplicate]

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $$ G\cong N \times G/N. $$ I tried to prove this claim, but then it seems that since $G$ is abelian then every ...
1
vote
0answers
23 views

Order of elements from coset and Schur-Zassenhaus Theorem (Exercise $3.B.6$ FGT)

I try to solve exercise 3.B.6 from M. Isaacs's book Finite Group Theory (FGT). I give it here. Let $N \vartriangleleft G$ and $g\in G$, and suppose that when $Ng$ is viewed as an element of $G/N$,...
0
votes
0answers
50 views

Normal subgroup and corresponding homomorphism

Topics in Algebra, a book written by Herstein, said the following thing on the first isomorphism theory: [The book says] Theorem 2.7.1 is important, for it tells us precisely what groups can be ...
4
votes
1answer
39 views

Show that a finite generated subgroup $H$ of $G$ with index 3 exists which is not normal in $G$.

> Suppose that $a$ & $b$ are the generators of a free group $G$.Show that a finite generated subgroup $H$ of $G$ with index 3 exists which is not normal in $G$. The way i try to solve the ...
2
votes
3answers
49 views

Finding a normal subgroup in $G := GL_2( \mathbb F_7)$ [closed]

For now, i know Sylow Groups and the theorem of structure for abelian groups. $G := GL_2( \mathbb F_7)$, $|G|=2^5 \cdot 3^2 \cdot 7 $ I m trying to show that: There exists a normal sub group in $G$...
0
votes
0answers
20 views

¨Show that all the square powers of every element are in the normal subgroup of index 2 [duplicate]

If $G$ is a group and $H$ is normal subgroup of index 2 in $G$, How to show that $x^2$ is in $H$ for all $x$ in $G$? Many thanks.
0
votes
1answer
50 views

covering spaces and equivalency of these three propositions [closed]

This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following ...
2
votes
0answers
27 views

Normal Subgroups and Quotient Groups help

Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$. (a) Prove that if $xH \in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$. (b) ...
0
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1answer
18 views

project normal subgroup generated by a subgroup to its abelianization

Say $\operatorname{Ab}(G)$ is the abelianization of $G$. Let $G_1$ and $G_2$ be two groups, $G_1\times G_2$ is the free product, then $G_1$ can be viewed as a subgroup in it. $j:G_1\times G_2\...
1
vote
1answer
39 views

Are normal subgroups of a profinite group with finite index closed?

Let $G$ be a profinite group and $H \subseteq G$ be a normal subgroup with $[G:H] < \infty$, i.e. $H$ has finite index. Question: Is $H$ closed? Problems: I have a lot of trouble to understand ...
0
votes
0answers
58 views

In a subgroup $H\le G$ of index $2$ every element $x\in G$ satisfies $x^2\in H$

I was thinking about this particular proof of the fact $$[G:H]=2\implies g^2\in H~\forall g\in G$$ 1) There is a bijection $f:G/H\mapsto H\backslash G$ defined by $f(xH)=Hx^{-1}$ It is well ...
1
vote
1answer
130 views

Let $G$ be a non-abelian finite $p$-group, do we have $Z(G) \leq \Phi(G)$ in general?

Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) \leq \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I ...
1
vote
0answers
32 views

Is G/H a group even if H is a subgroup of G but not a normal subgroup? [duplicate]

I know we can have left and right cosets without having the group be normal, but can we mod out by a subgroup without it being normal and still get a group? Thank you in advance!
3
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0answers
60 views

If $G$ has a nontrivial centre, must every subgroup of index $3$ be normal?

If a group $G$ has a nontrivial centre, must every subgroup of index $3$ be normal? $S_3$ yields an example of a group with a non-normal subgroup of index $3$, although it has a trivial centre. ...
1
vote
3answers
78 views

How is it possible that $\textrm{HK}=\textrm{G}$?

For the following problem: If $\textrm{H}$ and $\textrm{K}$ are distinct subgroups of $\textrm{G}$ of index $2$, then $\textrm{H}\cap\textrm{K}$ is a normal subgroup of $\textrm{G}$ of index $4$. ...
1
vote
2answers
44 views

Prove $H$ is a proper normal subgroup of $G$ if $H$ is generated by $\{[x,y] \mid x,y \in G\} \cup \{x^p \mid x \in G\}$.

I am trying to solve the following problem: Let $G$ be a nontrivial finite $p$-group, where $p$ is a prime, and let $H$ be the subgroup of $G$ generated by the set $\{[x,y] \mid x,y \in G\} \cup \{...
2
votes
2answers
32 views

Given that $G/H=\{xH:x \in G\}$ is group under operation $(xH)(yH)=(xyH)$ . Then $H$ is normal subgroup of $G$

let $G$ is any group and $H$ is its subgroup , such that $G/H=\{xH:x \in G\}$ is group under operation $(xH)(yH)=(xyH)$ . Then show that $H$ is normal subgroup of $G$ if $H$ is not normal ,then there ...
1
vote
1answer
31 views

Let $N$ be an Abelian normal subgroup of $G$, if $G/N$ is perfect, then also $G'$ is perfect.

I'm reading Kurzweil & Stellmacher's "The Theory of Finite Groups", its 1.5.3 says: Let $N$ be an Abelian normal subgroup of $G$. If $G/N$ is perfect, then also $G'$ is perfect. Proof. ...
-1
votes
1answer
53 views

Is there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$

Does there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$ My attempt: I think not, because for $n \ge 5$ , $A_n$ is the only normal subgroup of $...
-1
votes
1answer
93 views

Does normality of a subgroup imply it has index 2?

I know that if $G$ is a group, $N < G$, then the condition that $|G:N|=2$ implies that$ N$ is normal in $G$. But what about the converse if we know that $N$ is normal in $G$ does that then imply ...
1
vote
1answer
39 views

Show that $(M\times N)/R\cong M$.

If there are two groups, $M$ (with multiplication $\cdot_M$) and $N$ (with multiplication $\cdot_N$) and we define a new group $M \times N$ with multiplication such that $$ (m,n)(m',n') = (m \cdot_M ...
1
vote
1answer
54 views

Intersection of a Family of Normal Subgroups of a Group G is also a Normal Subgroup of G

I have this classic exercise to prove: If $ \{ N_i : i \in I \} $ is a family of normal subgroups of a group $ G $, then $ \bigcap_{i \in I} N_i $ is a normal subgroup of $ G $ too. I am searching ...
1
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0answers
30 views

Connected normal compact subgroups of $GL \left(2, \mathbb{C} \right)$

We know that any compact subgroup $G$ of $GL \left(2, \mathbb{C} \right)$ is conjugate to a closed subgroup of the $U(2)$ group. Since our group G is normal, it is simply a closed subgroup of the $U(2)...
2
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0answers
17 views

Index of a common normal core

Let $A, B, C$ be infinite groups and suppose that there are injective homomorphisms $\iota_A \colon C \to A$ and $\iota_B \colon C \to B$ such that $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. ...
4
votes
1answer
48 views

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent.

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent, where $\Phi(G)$ is the Frattini subgroup of $G$. The converse side ...
2
votes
0answers
26 views

Normal groups and homomorphism are the same, and this gives an approach to isomorphism theorem?

I was reading a post here that give some interesting approach about isomorphism theorem (see quote). But there are some things I don't understand. What exactly does this mean? The Second ...
11
votes
0answers
221 views

Are all verbal automorphisms inner power automorphisms?

Suppose $G$ is a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Lets call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, ...
1
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1answer
37 views

Normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially; What is wrong with the following trivial argument?

I'm trying to show that a normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially (please don't tell how to show it), but it seem it is quite a trivial question considering the following ...
2
votes
1answer
54 views

Does $G \times H = K$ iff $G$ and $H$ are normal in $K$? [closed]

Let $G \times H = K$. By applying The Isomorphism Theorem to the homomorphism $(g, h) \rightarrow g : K \rightarrow G$, I get $K/H \cong G$, so $H$ is normal in $K$. Similarly, $G$ is normal in $K$. ...
-1
votes
3answers
51 views

Suppose that $N$ is a normal and cyclic subgroup of $G$, and $H$ is a subgroup of $N$. Show that $H$ is normal in $G$. [closed]

Suppose that $N$ is a normal and cyclic subgroup of $G$, and $H$ is a subgroup of $N$. Show that $H$ is normal in $G$. What should I do when $N$ is finite? I have tried to show that it is right for ...
3
votes
1answer
53 views

$H$ is a subgroup of a finite group $G$ such that $|H|$ and $\big([G:H]-1\big)!$ are relatively prime. Prove that $H$ is normal in $G$.

Suppose that $H$ is a subgroup of a finite group $G$ and that $|H|$ and $\big([G:H]-1\big)!$ are relatively prime. Prove that $H$ is normal in G Let $[G:H]=m$ Let $G$ act on set $A$ of left cosets ...
0
votes
2answers
33 views

How do generators of a group work?

$G$ is a group, $H$ is a subgroup of $G$, and $[G:H]$ stands for the index of $H$ in $G$ in the following example: Let $G=S_3$, $H=\left<(1,2)\right>$. Then $[G:H]=3$. I know the definition of ...
1
vote
2answers
29 views

Suppose that $H\trianglelefteq G$ and $H\leq K\leq G$. Show $H\trianglelefteq K$ and that $K/H\leq G/H$

This is not for an assignment. I found this practice problem in the back of my abstract algebra book and I'm trying to figure it in preparation for an upcoming exam. I've attempted the first part of ...
1
vote
1answer
53 views

Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$. [closed]

Let $G$ be a finite group, $N\mathrel{\lhd}G$ a normal subgroup of $G$, and $H\leq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $\gcd(|H|,|N|)=1$). Show that the ...
0
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0answers
24 views

Normal Subgroup Representation Theory [duplicate]

Let $N \subset G$ be a normal subgroup. Show there exists a finite collection of irreducible representations $\phi_i: G \rightarrow GL(V_i)$ of $G$ such that $$ N = \bigcap{\ker(\phi_i)}$$
0
votes
1answer
30 views

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H).

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H). I saw a proof online that was pretty long, but can't I just argue that if $H \subset N(H)$, then $H$...
2
votes
1answer
39 views

Prove $G/(M\cap N) \cong M/(M\cap N) \times N/(M\cap N)$ where $G=MN$ and $M,N\triangleleft G$

If we consider the map $\phi: M\times N \rightarrow M/(M\cap N) \times N/(M\cap N)$, I was able to show that this is onto and the kernel of the map is $(M\cap N) \times (M\cap N)$ and hence by first ...
-1
votes
2answers
48 views

Is H a normal subgroup in G? [closed]

Let $G = S_5$ and let $H = \langle(1, 2, 3, 4, 5)\rangle$. Is $H$ a normal subgroup of $G$ ? Having some trouble figuring out this problem, it would be great if someone can help to find it!
0
votes
1answer
26 views

isomorphism of normal subgroup [closed]

I am reading group theory (particularly isomorphism) in the algebra, and stuck on a problem. Hope you guys will help me out: Let $G$ be finite group, and $A$,$B$ be normal subgroups of $G$ such that $...
2
votes
2answers
27 views

Proving an Intersection of Subgroups is Normal

Let $N$ be a subgroup of the group $G$. Show that $$X =\bigcap_{g \ \in \ G} \ g^{-1}Ng $$ is normal in $G$. It is easy to show $X$ is a subgroup in itself, by just showing $g^{-1}Ng $ is a ...
3
votes
1answer
40 views

Does the Lie group $G_2$ contain any normal subgroups?

A nonabelian Lie group is called a simple Lie group if it contains no nontrivial connected normal subgroups. On the other hand, a group is called simple if it contains no nontrivial normal subgroups. ...
0
votes
1answer
44 views

Normal subgroup and conjugate classes

My solution I am not sure if my solution is correct .
2
votes
1answer
25 views

Finding commutator subgroup $[D_3, D_3]$ of $D_3$ and showing $[D_3, D_3]$ is a normal subgroup of $D_3$

In my abstarct algebra class there is a two part question regarding $D_3$. The first part asks us to find a commutator subgroup $[D_3, D_3]$ of $D_3$. I am unsure on where to begin but I will state ...
1
vote
1answer
25 views

The group homomorphism $z \mapsto (z/|z|)^2$ with an application of the homomorphism theorem.

Question Let $G = (\mathbb{C}\setminus\{0\}, \cdot)$. 1) Show that $N = (\mathbb{R}\setminus \{0\},\cdot)$ is a normal subgroup. 2) Show that $f:G \rightarrow G$ with $z \mapsto (z/|z|)^2$ is a ...
0
votes
0answers
47 views

Let $G$ be a group and $H, H'$ be subgroups of $G$ where $H$ is normal. Under which circumstances is $H \cap H'$ a normal subgroup of $H$?

To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient? For instance, we could say $H \subset H'$, but ...
1
vote
3answers
131 views

Is it true that if $H_1$ and $H_2$ are isomorphic cyclic subgroups of $G$, then $G/H_1\cong G/H_2$?

I have a question: it is true that if $H_1$ and $H_2$ are cyclic groups that are isomorphic, then $G/H_1$ is isomorphic to $G/H_2$? I know that if I remove the condition "cyclic groups", the given ...
1
vote
2answers
54 views

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$. Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H \...
1
vote
1answer
55 views

If we divide a group and a proper subgroup of this group by the same normal subgroup, can the quotients be equal?

Let $G$ be a group and $H$ be a proper subgroup of $G$. Let $N$ be a normal subgroup of both $G$ and $H$. Question: Is it possible that $G/N = H/N$? Motivation: For a finite extension $L/K$ of ...
1
vote
1answer
52 views

The order of an element in a quotient group

Suppose $G$ is a finite group, that $H$ is a subgroup of $G$, and that $N$ is a normal subgroup of $G$. Suppose that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime. Consider the quotient ...