Questions concerning normal-subgroups of groups.

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2
votes
2answers
35 views

Group theory question for cyclic group

I met some problem during googling. The problem and its solution are next. and I'm wondering about YELLOW BOX $$ $$ $$ $$ Why $G$ has a unique element of order 2 in case of $H=G$ ? $$ $$ ...
-1
votes
0answers
21 views

Let G be a group and let x be a fixed element of G. Define Γ(x) = {g ∈ G : gx = xg} [on hold]

(a) Prove that Γ(x) is a subgroup of G. (b) Let G = A4, let x = (1 3)(2 4) and let y = (2 4 3). Find (i) Γ(x) and (ii) Γ(y).
0
votes
0answers
34 views

Sufficient conditions for $G\cong N\times G/N$ [duplicate]

Given a normal subgroup $N$ of a group $G$, do there exist sufficient conditions that allow us to conclude that we have an isomorphism $$ G\cong N\times G/N?$$
0
votes
2answers
37 views

Multiplication of subgroups

Let $H$ and $K$ be subgroups of a finite group $G$. Define $HK = \{hk\mid h \in H, k \in K\}$ and $KH = \{kh\mid k \in K, h \in H\}$. a) Show that in general $HK \ne KH$. (For example, consider $G = ...
0
votes
0answers
23 views

Automorphisms of $S(3,4,10)$

I'm trying to prove the next result: The group of automorphisms of the Steiner system $S(3,4,10)$ has a subgroup of index $2$ isomorphic to $S_6$. It'd be apprecciated if someone gave me a hand with ...
3
votes
1answer
53 views

Prove that : A group $G$ of order $p^n$ has normal subgroup of order $p^k$ , for all $0≤k≤n$

Prove that : A group $G$ of order $p^n$ has normal subgroup of order $p^k$ , for all $0≤k≤n$. I already appreciate your hints/answers
1
vote
0answers
16 views

groups and symmetry

Which of the following may be false? A) Any non-trivial element of $C_n$ generates $C_n$. B) Any subgroup of $C_n$ is cyclic. C) if $m|n$ then $C_n$ has at least one subgroup of order $m$. D) if ...
0
votes
1answer
30 views

Does existence of Index 2 s.g. imply existence of involution element?

I'm studying group theory (intro course) at the moment. Recently I made the error assuming that we can always construct an isomorphism between a quotient group and a subgroup. I've learned that this ...
1
vote
1answer
27 views

Special Linear group

Let $F$ be any field. Verify the group axioms for the special linear group $SL_n(F)$ whose elements are $n$ x $n$ invertible matricies with entries in $F$ and the product is matrix multiplication. ...
1
vote
1answer
42 views

Number of congruence relations of a 4-element non-cyclic group

How many congruence relations does a 4-element non-cyclic group have? Am I right that I have to find the normal subgroups in order to find the congruence relations? Thanks
0
votes
2answers
20 views

Maximal normal subgroup has prime index

I am trying to solve the following exercise taken from Rotman's An Introduction to the Theory of Groups: Let $M$ be a maximal subgroup of $G$. Prove that if $M \lhd G$, then $[G:M]$ is finite and ...
1
vote
1answer
36 views

Group of order $1575$ problem

Problem Let $G$ be a group with $|G|=1575$. If $H \lhd G$ and $|H|=9$, then $H \subset Z(G)$. What I've done so far is $|G|=1575=3^25^27$. I consider $G$ acting on $H$ by conjugation, or, in other ...
0
votes
0answers
27 views

Question about the third and fourth isomorphism theorems for groups

I am trying to work with both of the third and fourth isomorphism theorems for groups. I am considering the following situation: I wanted to take a subgroup in the quotient and see how it corresponds ...
-2
votes
0answers
39 views

A cyclic group $U_{34}$

For the elements of $U_{34}$ I know there are four subgroups. $U_{34}$ being the group generated where (a,m)=1 and m is 34. So $U_{34}$=$ [1,3,5,7,9,11,13,15,19,21,23,25,27,29,31,33]$. What are the ...
0
votes
1answer
25 views

Example of group with normal subgroup $N\ne\{e\}$ such that $N \cap Z(G)= \{ e\} $ and $G $ \ $ N$ contains an element of order more than $2$

(i) Give example of a group (if exists ) which has a normal subgroup $N\ne\{e\}$ such that $N \cap Z(G)= \{ e\} $ and $G $ \ $ N$ contains an element of order more than $2$ (ii) Give example of a ...
1
vote
2answers
38 views

Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
1
vote
1answer
37 views

$(\mathbb C^{\times}, \cdot)$ is a subgroup of $(GL(n,\mathbb C), \cdot)$

I am learning groups and subgroups in my algebra course. Today, we talked about examples of subgroup but I am not sure why the following holds: $(\mathbb C^{\times}, \cdot)$ is a subgroup of ...
1
vote
1answer
46 views

Normal subgroup and index problem

Let $G$ be a group and let $N$ be a normal subgroup of $G$ of finite index. Show that if $H$ is a finite subgroup of $G$ whose order is coprime with $[G:N]$, then $H$ is a subgroup of $N$. I don't ...
1
vote
1answer
46 views

Sylow subgroups problem

Let $G$ be a finite group and $p<q$ such that $p^2$ doesn't divide $|G|$. Let $H_p$ and $H_q$ be Sylow subgroups of $G$ with $H_p \lhd G$. Show $H_pH_q \lhd G \space \implies H_q \lhd G$. From the ...
0
votes
1answer
25 views

Is this set a generating set for this (normal) subgroup?

Suppose $G$ is a finite group and $N$ is a normal subgroup of $G$. Suppose also that $X$ is a generating set for $G$, and that $Y$ generates $N$ as a normal subgroup of $G$ (i.e. $N$=$\langle ...
1
vote
2answers
48 views

$N$ a normal subgroup of $G$ with $|G|$ odd and $|N|=5$ satisfies $N \subset Z(G)$

Exercise Let $N$ be a normal subgroup of $G$. Suppose that $|N|=5$ and that $|G|$ is odd. Show that $N \subset Z(G)$. I am sorry for not writing any work of mine but I really don't know where to ...
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votes
1answer
23 views

What is an example of a proper normal subgroup of the kernel of a homomorphism?

I'm reading this proof by Hungerford that concerns any normal subgroup of the kernel of a homomorphism. I understand the proof well enough, but I wanted to have some concrete example to guide or ...
1
vote
1answer
20 views

How to show $F(S)$ is normal in Sym$(S)$

As a follow-up of this question On non-trivial normal subgroup(s) of $A(S)$ , where $S$ is infinite , how do we show that the finitary symmetric group $F(S)$ is a subgroup and normal in Sym $(S)$ , ...
3
votes
1answer
28 views

On non-trivial normal subgroup(s) of $A(S)$ , where $S$ is infinite

Let $A(\mathbb R)$ be the permutation group of $\mathbb R$ , is this group simple ? In general for an infinite set $S$ , how may we determine whether $A(S)$ has any non-trivial normal subgroup or not ...
0
votes
1answer
52 views

Normal subgroup of a finite group

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?
0
votes
0answers
25 views

subgroups of the cyclic groups [duplicate]

Suppose a group G has no proper subgroups (that is, the only subgroups of G are G itself and the trivial subgroup {e}. Show that G is cyclic. With that may also get some example on this as well.
-2
votes
3answers
61 views

If $N$ normal subgroup of $G$ and $M$ normal subgroup of $G$ prove $MN$ is a subgroup [closed]

If $N\vartriangleleft G$ and $M\vartriangleleft G$ and $MN = \{mn | m \in M, n \in N\}$, prove that $MN$ is a subgroup of $G$ and that $MN\vartriangleleft G$
1
vote
0answers
20 views

How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up ...
3
votes
1answer
43 views

A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.

I want to prove the following: Let $G$ be a group of order $2^nm$, where $m$ is odd, having a cyclic Sylow $2$-subgroup. Then $G$ has a normal subgroup of order $m$. ATTEMPT: We ...
1
vote
2answers
39 views

Normal subgroup acting on a set

I am trying to solve the following problem: Let $G$ be a group acting on a set $X$ and let $S \lhd G$. Determine the necessary and sufficient conditions so that there exists an action of $G/S$ on $X$ ...
1
vote
2answers
69 views

Semidirect product, normal subgroup exercise

Let $G$ be a group and let $H,K$ be subgroups of $G$ such that $G=H \rtimes K$. (i) Show that if $K \lhd G$, then $kh=hk$ for all $h \in H, k \in K$. (ii) Deduce that $G$ is abelian if and only if ...
4
votes
2answers
45 views

$H$ of order $p$ normal in $G$ , g.c.d.$(|G|,p-1)=1$ , to prove that $H \subseteq Z(G)$

If $G$ is a finite group and $H$ is a normal subgroup of $G$ of order $p$(prime) such that g.c.d.$(|G|,p-1)=1$ , then how to prove that $H \subseteq Z(G)$ ? Please don't use any Sylow theorem or ...
2
votes
4answers
67 views

Calculate quotients of $\mathbb S_3$ and $\mathbb D_4$

Problem Calculate all the quotients by normal subgroups of $\mathbb S_3$ and $\mathbb D_4$,i.e., charactertize all the groups that can be obtained as quotients of the mentioned groups. For the case ...
0
votes
1answer
54 views

External semidirect product application

I am trying to find all normal subgroups of $\mathbb D_n$. I've read here Normal subgroups of dihedral groups that one could show the external semidirect product $(\mathbb Z/n\mathbb Z) \rtimes ...
1
vote
1answer
21 views

Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
1
vote
2answers
73 views

Proving certain groups are normal

Given the following subgroup of $\mathbb S_4$: $$K=\{\mathrm{id},(12)(34),(13)(24),(14)(23)\}$$ prove that $K \unlhd \mathbb A_4$, $K \unlhd \mathbb S_4$ I am trying to solve this problem doing as ...
2
votes
1answer
34 views

$G_i$ s are normal subgroups , then $\bigl[G:\bigcap_{i=1}^n G_i \bigr]\Bigm|\prod_{i=1}^n[G:G_i]$?

If $G_i$ $(i=1,\dots,n$) are normal subgroups of $G$, of finite index, then is it true that $\displaystyle\bigl[G:\bigcap_{i=1}^n G_i \bigr]\Bigm|\prod_{i=1}^n[G:G_i]$?
1
vote
1answer
35 views

A Group Isomorphic To The Direct Product Of Two of its Subgroups. Then are These Subgroups Normal?

Let $G_1$ and $G_2$ be subgroups of a group $G$. Assume that $G$ is isomorphic to $G_1\times G_2$. Then is it necessary that $G_1$ and $G_2$ are normal in $G$? Clealry $G_1\cong ...
1
vote
2answers
41 views

Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
0
votes
2answers
24 views

Isomorphism between quotient groups

Exercise Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$. As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying ...
3
votes
1answer
40 views

Problem on normal subgroups

Problem Let $G$ be a group and $H,K$ subgroups of $G$, we define $HK=\{h.k : h \in H, k \in K\}$. Prove that if $H$ or $K$ is normal, then $HK$ is a subgroup. In order to show $HK$ is a subgroup, ...
0
votes
2answers
38 views

$H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies $ $H$ is normal in $G$

Let $G$ be a group . If $H :=\{ g^2 : g \in G\}$ is a subgroup of $G$ , the how to prove that $H$ is normal in $G$ ?
3
votes
1answer
69 views

Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

An exercise from Dummit and Foote pg. $101$ ex. $9$ asks to show the following: Let $G$ be a group of order $p^{a}n$ where $p$ does not divide $n$ and let $N\unlhd G$ so that $|N|=p^{b}m$ where ...
1
vote
0answers
47 views

When $H \subsetneq N_G(H)$?

For which subgroups $H$ of a given group $G$ , is it true that $H$ is a proper subgroup of $N_G(H)$ , the normalizer of $H$ ?
3
votes
3answers
95 views

$H \lhd N \lhd G$ but $H \ntriangleleft G$ [duplicate]

I got stuck on the following exercise: Find a group $G$, a subgroup $N$, a subgroup of this subgroup $H$ such that $$ H \lhd N \lhd G \quad \text{but} \quad H \ntriangleleft G $$ This is what I ...
1
vote
2answers
44 views

Normal subgroup of a group

Let $H$ be a subgroup of $G$ and $K$ be a normal Subgroup of $G$. I need to prove that $KH$ is a subgroup of G where, $KH=\{\text{$kh$ : $k \in K \wedge h \in H$}\}$. Can somebody please help me in ...
2
votes
3answers
53 views

Define a normal subgroup of G

$N$ is a normal subgroup of $G$ if $aNa^{-1}$ is a subset of $N$ for all elements $a $ contained in $G$. Assume, $aNa^{-1} = \{ana^{-1}|n \in N\}$. Prove that in that case $aNa^{-1}= N.$ If $x$ is ...
1
vote
1answer
36 views

Proving $H$ is a normal subgroup of $G$ if $H$ and $gH$ ($g\notin H$) are the only distinct left cosets

Let $G$ be a group, and $H$ a subgroup of $G$. If $H$ and $gH$ where $g\notin H$are the only two distinct left cosets in $G$, prove that $H$ is a normal subgroup. I understand that $\{H, gH\}$ forms ...
1
vote
3answers
58 views

A subgroup such that at least one left coset is a right coset

I saw an exercise that goes: Let $G$ be a group of order 120, and $H$ be a subgroup of order 24. If at least one left coset of $H$ in $G$ is a right coset apart from $H$ itself, show that $H$ is ...
2
votes
1answer
40 views

Does this property of normal subgroups have a name?

I've recently started studying abstract algebra and I noticed that a subgroup N is normal iff $a \equiv x$ and $b \equiv y$ implies $ab \equiv xy$ mod N for any a,b,x,y. Is this an important ...