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Questions tagged [normal-subgroups]

For questions concerning normal subgroups of groups.

3
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0answers
38 views

The elements of a matrix group with order two and its centre

Let $$ G=\left\{ \begin{bmatrix} \bar{a} & \bar{b} \\ \bar{0} & \bar{c} \end{bmatrix} \text{with $\bar{a}$ and $\bar{c}$ in $\mathbb{F}^{*}_{7}$ and $\bar{b}$ in $\mathbb{F}_{7}$}\...
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2answers
55 views

If $aH=bH \implies Ha=Hb$ for a subgroup H having *finite index*, then $gH=Hg$ for all $g \in G$?

Problem 2.5.9 of Herstein's Topics in Algebra asks us to prove that if $H$ is a subgroup of a group $G$ such that $Ha \not = Hb \implies aH \not = bH$, then $gHg^{-1} \subset H$ for all $g \in G$, ...
1
vote
1answer
91 views

Locus equation in a non-simple group_part#2

(This is more a focused post than this similar one of mine, and the question more direct.) Let $G$ be a group and $H \triangleleft G$, $H \ne \lbrace e \rbrace$. For given $h \in H$ and $g \in G$, ...
1
vote
1answer
48 views

How to show that $C_G(a)=N_G(a)$?

For singleton sets, then how show to that $C_G(a)=N_G(a)$? As I know something about singleton set that is its closed in real line. Now I'm confused that how can I relate this concept in ...
0
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0answers
32 views

Group Homomorphisms into an Abelian Group

The following comes from Hungerford's Algebra. [Prove that if] $f: G \to H$ is a homomorphism, $H$ is abelian and $N$ is a subgroup of $G$ containing $\ker f$, then $N$ is normal in $G.$ A ...
3
votes
0answers
92 views

Locus equation in a non-simple group

Let $G$ be a group and $H \triangleleft G$, $H \ne \lbrace e \rbrace$. For given $h \in H$ and $g \in G$, let's set $R_g(h):=C_G(h)g \cap O_h$, where $C_G(h)$ is the centralizer of $h$ in $G$, and $...
2
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0answers
64 views

$H \triangleleft G$, $h \in H$. $C_G(h)g \cap H \ne \emptyset, \forall g \in G$ implies $\lbrace C_G(h)g \cap H, g \in G \rbrace$ partition of $H$?

Let $G$ be a group and $H \triangleleft G$. I know that $G$ fixes the conjugacy classes of $H$ under conjugation if and only if $\forall h \in H, C_G(h)g \cap H \ne \emptyset, \forall g \in G$. Now,...
0
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0answers
29 views

Deck Transformations Question

Using the Hatcher's notation on proposition 1.39. Let $G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$ be the Deck transformation group for the covering space $p: \tilde{...
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votes
2answers
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How do I Prove the direct product of two subgroups is a subgroup? [closed]

Let $P$ be a subgroup of the group $G$ and $Q$ be a subgroup of the group $H$. How would you Prove that: $P \times Q$ is a subgroup of the direct product $G \times H$.
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0answers
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Is there a way to classify all finite groups $G$, such that $\pi(G) := \Pi_{H \triangleleft G} |H| = |G|^2$?

Is there a way to classify all finite groups $G$, such that $\pi(G) := \Pi_{H \triangleleft G} |H| = |G|^2$? For abelian groups this problem is quite simple. All such abelian groups are exactly $C_{p^...
3
votes
2answers
49 views

$H/K$ when $K$ is not a subgroup of $H$.

Let $G$ be a group and $H$, $K$ ($K$ is normal) two subgroups of $G$ but neither $H$ is subgroup of $K$ nor $K$ is subgroup of $H$. What would be the problem with $H/K$? If I define for $x,y \in H$ ...
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1answer
37 views

Let $G$ be Abelian. Then any subgroup of $G$ is normal. Does the converse hold? [duplicate]

I need a little help with the following problem of abstract algebra: Let $G$ an Abelian group. Clearly, any subgroup of $G$ is normal. Is the opposite true, that is if every subgroup of $G$ is ...
2
votes
2answers
43 views

Prove that there is $x \in G$ such that $x \notin H, x^2 \notin H,…,x^{p - 1} \notin H$ but $x^p \in H$.

Full question: Let $p$ be a prime and let $k$ be a positive integer. Let $G$ be a group and let $H \triangleleft G$ with $[G : H] = p^k$. Prove that there is $x \in G$ such that $x \notin H, x^2 \...
1
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1answer
121 views

Visualization of groups with a normal subgroup

Suppose $G$ a group and $H \triangleleft G$ (proper normal subgroup). The simplest way to visualize this basic setup is that (Venn-wise) of a bubble ($H$) into a bigger one ($G$), sharing the unit and ...
3
votes
0answers
66 views

Group of square free order with a normal $p$-Sylow is solvable

Let $G$ be a group of order $p_1...p_s$ where $p_1,...,p_s$ are distinct primes. If $G$ has a normal $p$-Sylow subgroup, then $G$ is solvable. We proceed by induction on $s$. If $s = 1$, $G$ is ...
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0answers
25 views

Conjugation action on cosets

Given a group $G$ and a subgroup $H < G$, does conjugation given an action on cosets of $H$ via $g \cdot xH = gxHg^{-1}$? If it does it seems there is an easier proof to the problem in Normal ...
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0answers
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Visual representations of groups (in their symmetric groups)_part#2

Background In this post, I have shown that a plausible visual representation of a group $K$ in $\operatorname{Sym}(K)$ can be established, where $\operatorname{Aut}(K) \setminus \lbrace \iota_{\...
5
votes
1answer
235 views

Is every normal subgroup the kernel of some self-homomorphism? [duplicate]

Let $G$ be a group. If there is a homomorphism $f:G\to G$ (special case of the codomain being arbitrary group), then the kernel $f^{-1}(id)$ is a normal subgroup of $G$. But now the other way around: ...
3
votes
1answer
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$G$ group, $H \unlhd G$; $gC_G(h) \cap H \ne \emptyset,\forall h \in H,g \in G$ iff $G$ fixes the conjugacy classes of $H$ under conjugation.

I have learnt here that, given a group $G$ and $H \unlhd G$, it is $gC_G(h) \cap H \ne \emptyset,\forall h \in H,g \in G$, if and only if $G$ fixes the conjugacy classes of $H$ under conjugation. Then,...
4
votes
1answer
80 views

$G$ group, $H \unlhd G$; prove/disprove $gC_G(h) \cap H \ne \emptyset, \forall h \in H, g \in G$.

Let $G$ be a (possibly infinite) group and $H \unlhd G$. For a given $h \in H$, $G$ is partitioned into the set of (e.g. left) cosets of $C_G(h)$ in $G$. I would like to prove/disprove the following ...
2
votes
1answer
41 views

Abstract algebra: Normal subgroup problem

I got stuck on this specific problem: Let $G$ be a group and $H \mathrel{\unlhd} G$ with $|G : H| = p^k$ with $p$ is a prime and $ k \geq 1$. Prove that there exists a $g \in G$ such that $g ...
1
vote
1answer
32 views

Quotient group implies normality

We know that for a group $G$ and a normal subgroup $H \triangleleft G$, the operation $g_1Hg_2 H := g_1g_2H $ is well-defined and in fact results in a group structure on $G/H$. Conversely, I want to ...
8
votes
5answers
543 views

Almost normal subgroups: Is there any notion which is weaker than normal subgroup?

Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $\forall g \in G$, $g^{-1}Ng = N$. Is there any notion which is weaker than normal subgroup? I mean something ...
4
votes
1answer
70 views

Group of order $5^k\cdot 8$ has normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$

Let $G$ be a group of order $5^k\cdot 8$. I was trying to prove that there are normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$. I saw the following statement: Let $P$ be a $p$-Sylow ...
1
vote
3answers
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Let $N,M$ be normal subgroups of $G$ with $N\cap M=\{e\}$. Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$.

First consider the following definition: Let $G$ be a group and $H$ a subgroup of $G$. The center: $$ C_{G}(H)=\{g\in G\,:\,gh=hg,\,\forall h\in H\}$$ Now I'm trying to prove the following ...
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1answer
37 views

How to prove this property of this group of order $20$ without the Sylow theorems?

In Artin's Algebra under the section on the Class Equation is the exercise The class equation of a group $G$ is $1+4+5+5+5$. (a) Does $G$ have a subgroup of order $5$? If so, is it normal? (b) Does ...
0
votes
2answers
65 views

Let $H$ be a subgroup of a group $G$. Show that for $a,b\in G$ we have $aH=bH$ if and only if $a^{-1}b\in H$

Group $H \leq G$. Then for $a,b\in G$, $aH=bH \iff a^{-1}b\in H$ My Proof: ($\Rightarrow$) Suppose $aH = bH$, then $a^{-1}aH=a^{-1}bH \ldots e = ab^{-1} \in H$ How to I prove conversely ($\...
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3answers
72 views

How do I know that $\mathbb{Z}_{175}$ is not an additional subgroup of order $175$?

Here was the original problem statement. Enumerate all non-isomorphic groups of order $175$. See that $|G| = 175 = 5^2\cdot7$. Therefore by Sylow's first $H \leq G$ & $|H| = 25$ in addition to ...
4
votes
2answers
86 views

Is the normality of a subgroup dependent on which group is its parent?

It is important to understand the relationship of normal subgroups to their parent. One concept that needs to be understood is whether the normality of a subgroup does not depend on which parent group ...
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0answers
38 views

Proving $(U \cap V) \cap u(U \cap v) = (u \cap V)(U \cap v)$

Let $U, V$ be subgroups of a group. Let $u \trianglelefteq U, v \trianglelefteq V$. I proved like this. Then by applying modular law, $$\mathrm{(LHS)} = (U \cap V) \cap u(U \cap v) = U \cap V \cap uv \...
1
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1answer
33 views

Property of Sylow $2$-subgroups of $S_n$ and $S_{n+1}$

The question Let $P$ be a Sylow $2$-subgroup of $S_n$ and let $Q$ be a Sylow $2$-subgroup of $S_{n+1}$. Show $Q\cong P\times C_2$ iff $n\equiv 1\pmod{4}$. My attempt The first direction is ...
2
votes
2answers
40 views

Excercise with the normal subgroup $K$ of $G$ with $K:=\bigcap\{H \text{ subgroup of } G: \forall_{x,y\in G}:xyx^{-1}y^{-1}\in H\}$

So far I have showed that $K$ is a normal subgroup and that the Operation defined on $G/K$ is abelian. Now I have to show that if $\phi:G\rightarrow A$ is a group homomorphism and $A$ is abelian....
2
votes
1answer
44 views

How do the elements of $(G\times G) /D$ look like? $D:=\{(g,g)\in G\times G\}$ and $G$ is abelian

The first Thing I have done was to verify that $D$ is a normal subgroup. I did so by verifying that $$f:G\times G \rightarrow G, f(a,b)=a^{-1}b$$ is linear, and that the kernel of $f$ is $D$. How ...
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votes
1answer
36 views

The groups $G_1, G_2$ and the normal subgroups $N_1,N_2$ are isomorphic. Dis/prove $G_1/N_1\cong G_2/N_2$ [duplicate]

Let $G_1, G_2$ be groups, and let $N_1\trianglelefteq G_1, N_2\trianglelefteq G_2$ such that $G_1\cong G_2, N_1\cong N_2$. Prove or disprove $G_1/N_1\cong G_2/N_2$. Let $\varphi:G_1\to G_2, \psi:N_1\...
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votes
1answer
34 views

If N is a normal subgroup of a group G, then prove that G/N is a group with respect to multiplication of cosets. [closed]

I can prove that G/N follows closure property, associative property, has identity and inverse with respect to multiplication of cosets. But after this, what should I do to prove that G/N is a group ...
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votes
1answer
25 views

Groups and normal subgroups

Let H be normal subgroup of G, and H∩G'={e} , G'={[x,y]=xyx^(-1)y^(-1) |x,y∈G}. Prove that ∀g ∈ G and ∀h∈ H gh=hg.
2
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1answer
64 views

If $H$ is a subgroup, and $xHx^{-1} \subsetneq H$, for all $x$ in $G$, then is H a normal subgroup?

Here $xHx^{-1} \subsetneq H$ means $xHx^{-1}$ as to be a proper subset of $H$. From Gallian, Contemporary Abstract Algebra: Normal Subgroup: A subgroup $H$ of a group $G$ is called a normal ...
0
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1answer
27 views

Characteristic subgroups and factorization

Let $G$ be a finite group, $C$ and $N$ be characteristic subgroups of $G$. Is it true that $CN/N$ is a characteristic subgroup of $G/N$? I try to answer this question in the case, where $N$ is a ...
3
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2answers
78 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
96 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 ...
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0answers
24 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 $G/N$,...
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0answers
56 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
42 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
51 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
53 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
28 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
votes
1answer
22 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
72 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 ...