Questions concerning normal-subgroups of groups.

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3
votes
2answers
71 views

$x=x^2$ in a sub group?

I have a set E defined in ℝXℝ (E=ℝXℝ) and the operation * defined like this ...
-2
votes
1answer
36 views

For a group G define the set $Z(G)$ by [closed]

$Z(G) = \{ fz \in G\mid zg = gz \;\forall g \in G\}$ . In other words, Z(G) is the set of all elements that commute with every other element. Show that Z(G) is an abelian subgroup of G I understand ...
2
votes
1answer
36 views

Check whether two conjugate subgroups are still conjugate in some subgroup

Given a group $G$ and $K<G$. Let $H_1,H_2$ be subgroups in $G$ satisfying: \begin{align} &H_1\triangleleft K;\\ &H_2\triangleleft K;\\ &H_1\sim_G H_2.\quad\mbox{(They are ...
3
votes
1answer
51 views

let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$ [duplicate]

Let $G$ be a finite group, $p$ be the smallest prime divisor of $|G|$ and x $\in$ G an element of order $p$. Suppose $ h \in G $ is such that $h^{-1}xh = x^{10}$. Show that $p = 3$. I cant ...
0
votes
0answers
21 views

Why do we need normal subgroup? [duplicate]

Thanks for reading this question. I'm new to abstract algebra. Up to now, I have learned the concept/definition and some corollaries w.r.t the normal subgroup, still I am not quite familiar with the ...
0
votes
3answers
42 views

True or not: If a normal subgroup and its quotient are commutative, then the group is.

Let $G,*$ be a group and $A,*$ a normal subgroup of $G,*$: $$ A \triangleleft G \quad( \equiv \forall g\in G: gA = Ag) $$ Then $G$ is commutative iff A and $\frac{G}{A}$ are commutative. I can see ...
2
votes
0answers
52 views

Prove there is $\sigma\in S_3$ such that $H_{\sigma (i)} \cong\textrm{}K_i ,\space \forall i $

In class they gave us a problem, After spending a long time trying to solve it, I turn to you =] Let $H_1,H_2,H_3, K_1,K_2,K_3 \le G$ be simple groups, $G=\{{h_1}{h_2}{h_3}:h_1\in H_1,h_2\in ...
1
vote
1answer
51 views

Operation with normal subgroup

I am working on a problem on finite group theory, and would like asking a question on the correct operation of normal subgroup. Suppose that $H$ is normal subgroup of $G$ and the factor group $G/H$ ...
1
vote
1answer
32 views

Does N/H=K/H under some terms mean N=K?

Let $G$ be a group and let $K$ be a normal subgroup of $G.$ Now let $H$ and $N$ be normal subgroups of $G$ containing $K.$ Given $N/K=H/K$ can I show $N=H$ necessarily? Is there a way?
0
votes
1answer
58 views

What is the composition series of $\mathbb Z_7$ x $\mathbb Z_{12}$

So I get the answer as follows (which is correct I believe): {$0$} x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x <$4$> $\vartriangleleft$ $\mathbb Z_7$ x ...
0
votes
0answers
16 views

P-groups in group theory/ normal groups [duplicate]

Let G be a finite p-group for some prime p. Prove that the number of subgroups of G that are not normal is divisible by p.
4
votes
2answers
49 views

Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$

Given: Let $G$ be a group, and let $\mathcal{S}$ be the set of subgroups of $G$. For $g\in G$ and $H\in S$, let $g\cdot H=gHg^{-1}$ Question: Deduce that if $G$ is a finite $p$-group, for some prime ...
2
votes
0answers
19 views

Show the map from the product of disjoint subgroups of G to G is an isomorphism [duplicate]

Question: Show that if $G$ is a group with two normal subgroups $H$ and $K$ such that $G=HK$ and $H\cap K=\{e\}$, then the map $(h,k)\mapsto hk$ is an isomorphism of groups from $H\times K$ to $G$ ...
2
votes
1answer
64 views

$h$ is non-identity element whose square is identity $\implies\langle h\rangle$ is normal

$h$ is non-identity element whose square is identity $\implies\langle h\rangle$ is normal If the whole group is $G$ and $h\in G$ with the property above then $\langle h\rangle=\{1,h\}$. Clearly ...
1
vote
2answers
44 views

Number of $p$-local subgroups of a group

A subgroup of a finite group is '$p$-local' if it is the normalizer of some Sylow $p$-subgroup. I want to prove that the number of $p$-local subgroups of a group is congruent to $1$ modulo $p$. I know ...
2
votes
2answers
44 views

$G$ contains at least $r(p-1)$ elements of order $p$

Suppose a group $G$ has $r$ distinct subgroups of prime order $p$. Show that $G$ contains at least $r(p-1)$ elements of order $p$. Aside: I know how to use this to prove that a group of order $56$ ...
3
votes
1answer
45 views

HK is cyclic if H and K are both cyclic

Let $G$ be a finite group with normal subgroups $H$ and $K$ of relatively prime orders. Show that the group $HK$ is cyclic if $H$ and $K$ are both cyclic. My attempt was to use the $2$nd Isomorphism ...
1
vote
0answers
31 views

A question on the symmetric commutator product

Let $G$ be a group and let $R_1,\cdots, R_n$ be subgroups of $G$, where $n\geq 2$. The symmetric commutator product of $R_1,\cdots, R_n$, denoted by $[R_1,\cdots,R_n]_S$, is defined as ...
0
votes
0answers
24 views

Terminology: how do people call the “normal generating set”?

Let $G$ be a group and let $x_1,\cdots,x_n\in G$ and let $A$ be the normal closure of $\{x_1,\cdots,x_n\}$; that is, the smallest (by inclusion) normal subgroup containing $\{x_1,\cdots,x_n\}$. Notice ...
2
votes
1answer
46 views

Size of the orbits of a normal subgroup

So this is the question: Let $H$ be a finite subgroup of $G$, and let $(h,h')(x)=hxh^{-1}$ define an achtion of $H\times H$ on $G$, prove that $H$ is a normal subgroup of $G$ if and only if every ...
3
votes
0answers
57 views

A name for the automorphisms induced by the normalizer by conjugation?

Let $N$ be a subgroup of a group $G$, $H := \operatorname{N}\left( N \right)$ the normalizer of $N$. There is a natural morphism from $H$ to $\operatorname{Aut}\left( N \right)$ given by $h \mapsto ...
0
votes
1answer
25 views

Show there exists a subgroup of order 15

I have a group $|G|=375=5^3*3$ by Sylow analysis, I have shown that $H_5$ is normal, but $H_3$ is not necessarily normal. My question is if I assume $H_3$ is normal, how do I show there is a subgroup ...
2
votes
1answer
29 views

Show this function is onto

I gave a mapping A to C such that A is the set of left cosets in G described as $A$={$N(H), gN(H),...g_nN(H)$} for N(H) is the normalizer of H in G and C is the set of conjugates of H, ...
0
votes
0answers
44 views

Show for $x\in G$ written as product of $hk$.

Show for $x\in G$ written as product of $hk$ for $h \in H$ and $k\in K$. Let $G$ be of order $p^km$ for $p$ is prime and does not divide $m$. $H=(x\in G\mid x^{p^k}=e)$ and $K=(x\mid x^m=e)$. G is not ...
3
votes
2answers
76 views

If $H\cap K=e$ then $H$ and $K$ are normal

For $|G|=p^km$ for $p$ is prime and $p$ does not divide $m$. Let $H=[x\in G \mid x^{p^k}=e]$ for $H<G$ and let $K=[x\mid x^m=e] $ for $K<G$. We are not assuming G is abelian Show that $H\cap ...
2
votes
1answer
36 views

Show $A_6$ is simple

I have to show that the group $A_6$ is simple. For the subgroups that have order divisible by $5$ and order of $8,9,18,24,36,$ and $72$ I have shown that those subgroups are not normal. Now we need to ...
0
votes
2answers
50 views

abstract algebra question with p-subgroup

Let $G$ be a finite group, $H<G$ a subgroup. Suppose that $H$ is a $p$-group, $p$ a prime, and $p\mid[G:H]$. Show that $p\mid[N_G(H):H]$ and $H<N_G(H)$. In particular, if $G$ is a $p$-group and ...
4
votes
3answers
155 views

Prove a quotient group is abelian

Let $G$ be a group with a normal subgroup $M$ such that $G/M$ is abelian. Let $N\geq M$ and $N \unlhd G$. Show $G/N$ is abelian. My attempt: To show that $G/N$ is abelian, we need to show that for ...
1
vote
0answers
27 views

Show the intersection of a nonidentity normal subgroup and the center of P is not trivial

P is p-group and M is a nontrivial normal subgroup of P. Show the intersection of M and the center of P is nontrivial. By the class equation, I proved that Z(P)is not 1. Then, how do prove I the ...
0
votes
1answer
47 views

If $N \lhd G, (G : N) = 100, a \in G, a^{23} = e$, show that $a \in N$.

I'm having some trouble with a question that my instructor suggested to think about: Let $N \lhd G$ be a normal subgroup of $G$, the index $(G : N) = 100, a \in G, a^{23} = e$. Show that $a \in N$. ...
0
votes
3answers
71 views

show that there exist an element $g$ of a group $G$ such that $g^q$ is in $H$

$H$ is normal in $G$ and $q$ is prime with $q\mid[G:H]$. Show that there exist $g$ in $G$ such that $g^{q}$ is in $H$. I am not sure how to use $q\mid[G:H]$. Should I try to show that $f:G→H$, with ...
1
vote
3answers
37 views

let $G$ be dihedral group so that $D_{2n}= \langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. Prove that $N\unlhd G$ and $G/N \cong W = \{1,-1\}$

let $G$ be dihedral group so that $G = D_{2n}=\langle x,y \ | \ x^2=y^n=e, yx=xy^{n-1} \rangle$. 1) let $N < G$ so that $N =\langle y \rangle = \{e,y,...y^{n-1}\}$. Prove that $G \unlhd N$. 2) ...
2
votes
2answers
50 views

A group in between the commutator subgroup and the original group must be normal

Let $C$ be the commutator subgroup of a group $G$. then by some easy arguments, we know that $1$. $C$ is normal in $G$ $2$. $G/C$ is abelian $3$. If $N$ is normal in $G$ and $G/N$ is abelian, then ...
0
votes
0answers
20 views

A question on normality of $\langle x \rangle Z(G)$ for non-abelian group $G$

Let $G$ be a non-abelian group , then I have noticed that for every $x \in G$ \ $Z(G)$ , $\langle x \rangle Z(G)$ is a subgroup such that $Z(G) \subset \langle x \rangle Z(G) \subset G$ ; I would ...
0
votes
1answer
31 views

Prove that H is a normal subgroup if and only if, $\forall a , b \in G, ab \in$ H implies $ba\in H.$

Let H be a subgroup of G. Prove that H is a normal subgroup if and only if, $\forall a , b \in G, ab \in$ H implies $ba \in H.$ I don't have a problem to prove one of the implications, however, this ...
1
vote
3answers
51 views

Let $G$ be a group. If $H\leq G$ is a subgroup and $N\vartriangleleft G$, then $HN$ is a subgroup of $G$.

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. Prove that $HN=\{hn\:|\:h \in H, n \in N\}$ is a subgroup. Here what I have so far. It is not really much I understand what I need to ...
0
votes
0answers
51 views

Find positive integer d of the additive group Z?

Let $H$ be a subgroup of the additive group $\mathbb{Z}$ given by $$H = \{70x-124y \mid x,y \in Z\}.$$ Find positive integer $d$ such that: $H = \{nd\mid n \in \mathbb{Z}\}$ Thanks in advance. ...
0
votes
1answer
33 views

Normal Subgroups in Group Theory

I am quite confused about the Group Theory. In particular, would like to ask whether is this statement true. If G is a group, and H is a normal subgroup of G, then |H| * |G/H| = |G| Thanks! Also, ...
0
votes
1answer
34 views

How to prove a subgroup is normal?

Prove that $D$ is a normal subgroup of $C$ if $C=S_3 \times \Bbb Z_4$ and $D=\langle((132),2)\rangle$. I know to prove a subgroup is normal you have to show aH=Ha, but I'm just not sure how to do this ...
2
votes
0answers
42 views

A Question on the Quotient Group and/or set of cosets

I'm just confused about a somewhat simple fact about quotient groups. If we have: $$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we ...
0
votes
1answer
48 views

Suppose $G$ is a group, with a subgroup $K$ and a normal subgroup $H$.

Let $h, h_1 \in H$ and $k, k_1 \in K$. If $hk = h_1k_1$, show that $h_1=hb$ and $k_1=b^{-1}k$, for some $b \in H \cap K$. I noticed that $H \cap K$ is a subgroup of $K$, and $HK$ is a subgroup of ...
0
votes
1answer
23 views

Testing Normality in a Group

Let H be a normal subgroup of G. Assume that $ab \in H$. Show $ba\in H$. Consider the conjugate of $ab\in H$ with the element $b$. My question is what is the conjugate of a $ab$ with $b$? Also the ...
0
votes
1answer
21 views

The converse of Factor Group criterion

If N is a normal subgroup of G then G/N is a group. Is the converse true? I mean If G/N is a group then N is Normal.
0
votes
1answer
41 views

Equality of cosets implies equality of the original sets

Let $H_1$, $H_2$ be two subgroups of $G$ containing $K$, where $K$ is a normal subgroup of $G$. Then if $H_1/K = H_2/K$, prove that $H_1=H_2$. Attempt: Let $h_1K = h_2K$, for some $h_1 \in H_1$, ...
4
votes
1answer
60 views

every group of order $105$ has a cyclic normal subgroup of index $3$ ?

Does every group of order $105$ has a cyclic normal subgroup of index $3$ ? (Please don't use Sylow theorems )
0
votes
2answers
30 views

Direct Product of Normal Subgroups is a Normal Subgroup of a Direct Product [closed]

If $A_1 \trianglelefteq G_1 $, $A_2 \trianglelefteq G_2$ then $\Rightarrow A_1 \times A_2 \trianglelefteq G_1 \times G_2$. Is this true? If so, how do I prove it?
1
vote
2answers
37 views

Homomorphism preserves normality

let $\phi:G\rightarrow G'$ be a homomorphism, and let N' be a normal subgroup of G'. I want to show that $\phi^{-1}[N']$ is also normal subgroup of G. My work : since homomorphism preserves ...
-1
votes
1answer
43 views

Alternate Form of Correspondence Principle

Let G be a group and K a normal subgroup in G. Let f : G → G/K be the canonical epimorphism given by x 􏰁→ xK and L a subgroup of G/K. Then (1) There exists a subgroup H of G. H contains K, and ...
1
vote
1answer
22 views

Show that a certain normal subgroup or a product is abelian

Let $A$ be a normal subgroup of $G\times H$. If the identity of $G$ is $1_G$ and the identity of $H$ is $1_H$, $(x,y)\in A$ has the property that $x\ne1_G$ and $y\ne1_H$ unless $(x,y)=(1_G,1_H)=1_A$. ...
0
votes
0answers
17 views

Normal series and refinements

I am trying to show that no nontrivial refinements exists in the following series but haven't been able to make any progress: $\{1\}\unlhd Z(SL_2(\mathbb{R}))\unlhd SL_2(\mathbb{R})\unlhd ...