Questions concerning normal-subgroups of groups.

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2
votes
2answers
32 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
52 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
23 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
28 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
39 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
73 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
34 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
0answers
19 views

To show that $S_4$ has a normal subgroup of order $4$ [duplicate]

WHow to show that $S_4$ has a normal subgroup of order $4$ ? . Please help .Thanks in advance
0
votes
2answers
48 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 ...
-1
votes
0answers
36 views

isomorphism and circle group [closed]

For my mathimatics study, I have to prove this: Let $U_{1}$ be the circle group on $\mathbb{C}^\times$. Prove that $\mathbb{C}^\times$ is isomorphistic to $U_{1}\times \mathbb{R}^+$. How do I ...
3
votes
3answers
141 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
23 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
41 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
69 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
30 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
46 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
18 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
27 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
46 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
49 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
31 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
33 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
39 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
42 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
20 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
37 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
56 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
29 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
35 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
39 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
16 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 ...
2
votes
1answer
198 views

Show that SO(n) is a normal subgroup of O(n)

Show that SO(n) is a normal subgroup of O(n). A normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. SO(n) is the set of orthogonal ...
1
vote
1answer
81 views

proof of a useful counting result in group theory

Let G be a finite group, H a subgroup of G satisfying |G| |̸| [G : H]!. Prove there exists a normal subgroup N of G satisfying 1 < N ⊂ H. maybe the General Cayley's Theorem works. I am not sure. ...
1
vote
2answers
57 views

The Normalizer and Centralizer of Groups

recently my teacher has introduced us two new Theorems called the N/C Theorem and the Normalizer Theorem. N/C states: Let $H<G$ and $N(H)$ be a normalizer of H in G. Let $C(H)$ be the centralizer ...
1
vote
3answers
36 views

Show that $a^m$ is in $H$ for every $a$ in $G$ [duplicate]

Let $H$ be a normal subgroup of $G$, and let $m=(G:H)$. Show that $a^m \in H$ for every $a \in G$. I have been thinking about this question for a few days but I get something informal. What am ...
1
vote
2answers
81 views

Let $G$ be a group and $H$ a normal subgroup of $G$. Prove: $x^2 \in H$ for every $x \in G$ iff every element of $G/H$ is its own inverse.

Let $G$ be a group and $H$ a normal subgroup of $G$. Prove: $x^2 \in H$ for every $x \in G$ iff every element of $G/H$ is its own inverse. Here is my proof. I've only tried proving one way ...
2
votes
2answers
56 views

Show that a subgroup $K$ is normal [duplicate]

Let $K$ be the subgroup $K=\{e,(1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ in $S_4$. show that $K$ is normal in $S_4$. show that $S_4/K \cong S_3$. I know that I could try to prove this directly ...
0
votes
0answers
25 views

Showing $A_5$ is simple

For $H$ is a normal subgroup of $A_5$. I know that the order of |aH| must divide both |$A_5$/H| and |a|. (Not entirely sure why it divides |a|, but that might be something to explain). From this we ...
2
votes
2answers
39 views

Coset Multiplication and well-defined.

Let H be a normal subgroup of G. $1$. Assume that H is a normal subgroup of G. Show the product $aHbH$ is independent of the representative chosen from $aH$ and $bH$, so show that $(ah)(bh') \in ...
1
vote
2answers
28 views

Proof that this is a normal subgroup

Let $\phi: G $ to $J$ be a homomorphism onto all of $J$. Let $H$ be a normal subgroup of $G$ and let $K=\phi(H)$ be the image of $H$ under the homomorphism. $1$. Show $K$ is a subgroup of $J$. $2$. ...
2
votes
1answer
15 views

Proof that, if factor groups are equal, then the subgroups are equal

I'm reading a proof of the Correspondence Theorem and I'm seeing a step for which I'm not sure of the justification. So suppose that $G$ is a subgroup and $K< H_{1},H_{2} <G$, and ...
1
vote
2answers
35 views

Consider $f: \mathbb Z_8 \rightarrow \mathbb Z_4$. List the cosets of the kernal of $f$

Consider $f: \mathbb Z_8 \rightarrow \mathbb Z_4$ given by $$\begin{pmatrix} 0&1&2&3&4&5&6&7 \\ 0&1&2&3&0&1&2&3 \end{pmatrix}$$ Verify the ...
2
votes
2answers
79 views

Properties possessed by $H , G/H$ but not G

i) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are abelian but $G$ is not ? ii) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are cyclic ...
0
votes
4answers
47 views

T/F questions on $p$-Sylow subgroups and normalizer

Prove the following statement if it is true. Otherwise, disprove it by giving a counterexample. 1) The normalizer $N$ in a finite group $G$ of a subgroup $H$ of $G$ is always a normal subgroup ...
4
votes
6answers
91 views

If the normalizer of a subgroup in a group is equal to the subgroup then is the subgroup abelian?

If $H$ is a proper subgroup of $G$ such that $H=N_G(H)$ ( the normalizer of $H$ in $G$ ) , then is it true that $H$ is abelian ?
0
votes
2answers
44 views

How is $S_n$ not simple for $n \geq 3$?

I know the only normal subgroups of $S_n, n \geq 3$, are itself and the trivial subgroup. But why is this the case?
0
votes
2answers
31 views

Largest Normal Subgroup of a Group G

Let $G$ be a group and let $Z(G)$ be the center of $G$. We know that $Z(G) \unlhd G$, but does that mean that $Z(G)$ is the largest normal subgroup of $G$?
2
votes
0answers
59 views

Show that $G/H\cong S_3$

If $G:=S_4$ and $H:=\{id,(12)(34),(13)(24),(14)(23)\}$ Show that $G/H$ has order $6$ and all of its elements have order less than or equal to $3$ (so by the classification of the groups of order 6 ...