Questions tagged [normal-subgroups]
For questions concerning normal subgroups of groups.
3
votes
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}$}\...
1
vote
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
votes
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
votes
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
votes
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{...
-3
votes
2answers
27 views
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$.
6
votes
0answers
86 views
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$ ...
-1
votes
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
vote
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 ...
0
votes
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 ...
1
vote
0answers
75 views
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
79 views
$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
30 views
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\}$$
Now I'm trying to prove the following ...
1
vote
1answer
37 views
How to prove this property of this group of order $20$ without the Sylow theorems?
In Artin's Algebra under the section on the Class Equation is the exercise
The class equation of a group $G$ is $1+4+5+5+5$. (a) Does $G$ have a subgroup of order $5$? If so, is it normal? (b) Does ...
0
votes
2answers
65 views
Let $H$ be a subgroup of a group $G$. Show that for $a,b\in G$ we have $aH=bH$ if and only if $a^{-1}b\in H$
Group $H \leq G$. Then for $a,b\in G$, $aH=bH \iff a^{-1}b\in H$
My Proof:
($\Rightarrow$)
Suppose $aH = bH$, then
$a^{-1}aH=a^{-1}bH \ldots e = ab^{-1} \in H$
How to I prove conversely ($\...
1
vote
3answers
72 views
How do I know that $\mathbb{Z}_{175}$ is not an additional subgroup of order $175$?
Here was the original problem statement.
Enumerate all non-isomorphic groups of order $175$.
See that $|G| = 175 = 5^2\cdot7$. Therefore by Sylow's first $H \leq G$ & $|H| = 25$ in addition to ...
4
votes
2answers
86 views
Is the normality of a subgroup dependent on which group is its parent?
It is important to understand the relationship of normal subgroups to their parent. One concept that needs to be understood is whether the normality of a subgroup does not depend on which parent group ...
1
vote
0answers
38 views
Proving $(U \cap V) \cap u(U \cap v) = (u \cap V)(U \cap v)$
Let $U, V$ be subgroups of a group. Let $u \trianglelefteq U, v \trianglelefteq V$. I proved like this. Then by applying modular law,
$$\mathrm{(LHS)} = (U \cap V) \cap u(U \cap v) = U \cap V \cap uv \...
1
vote
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 ...
0
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\...
0
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 ...
-4
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
votes
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
votes
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
votes
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 ...
1
vote
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 $G/N$,...
0
votes
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 ...
