Questions concerning normal-subgroups of groups.

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2answers
26 views

Necessary and Sufficient conditions to be a subgroup and/or a normal subgroup.

If $x \in G$, is it possible that $C = \{g^{-1}xg : g \in G \}$ is a subgroup of $G$? Can $C$ be a normal subgroup of $G$? (What are necessary and sufficient conditions to be such a subgroup?) ...
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0answers
35 views

$G/N\cong G/N'$ implies $N\cong N'$ for the normal subgroups? [duplicate]

We have normal subgroups $ N$ and $ N'$ of a group $ G $. Is the following statement correct? $G/N\cong G/N'\Leftrightarrow N\cong N'$. $\Leftarrow $ is trivial. $\Rightarrow:$ I don't know how to ...
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0answers
17 views

Proof of a more general correspondence theorem

I'm trying to prove the following, but I don't really know where to start... If $\theta : G \rightarrow H$ is a surjective groupmorfism with $ker(\theta) = N$. If we define$ S := \{U|N\leq U \leq ...
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2answers
43 views

prove that $(n \Bbb Z /k \Bbb Z)/(m \Bbb Z /k \Bbb Z) \cong n\Bbb Z / m \Bbb Z$

I need to prove thae following - given $n,m,k \in \Bbb N$ such that $n|m , m|k$ prove that $(n \Bbb Z /k \Bbb Z)/(m \Bbb Z /k \Bbb Z) \cong \Bbb Z / \frac mn \Bbb Z$ What I tried and what missing ...
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2answers
30 views

Normal subgroup in S4 [duplicate]

Let H be a subgroup of S4 where $H = \{e, B , C ,D \}$ $B(1)=2,B(2)=1,B(3)=4,B(4)=3$ $C(1)=3,C(2)=4,C(3)=1,C(4)=2$ $D(1)=4,D(2)=3,D(3)=2,D(4)=1$ Prove that H is a normal subgroup. I've tried ...
3
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3answers
71 views

Is every normal subgroup of a finitely generated free group a normal closure of a finite set?

Let $G$ be a group, $S\subset G$ a subset, then the smallest normal subgroup of $G$ which contains $S$ is called the normal closure of $S$, and denoted by $S^G$. My question is, if $G$ is a free ...
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3answers
39 views

Given a group $G$ and subgroup $H<G$ and $g \in G$ such that $gHg^{-1} \subset H$ prove

My problem is the following - Given a group $G$ and subgroup $H<G$ and $g \in G$ such that $gHg^{-1}\subset H$ prove that if H is a finite group $|H|< \infty$ then $gHg^{-1} = H$ what I tried ...
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1answer
39 views

$G$ is a group and $N,M$ are normal subgroups of $G$. Prove that $nm=mn$ for all $n\in N,m\in M$. [duplicate]

My problem is the following $G$ is a group and $N,M$ are normal subgroups of $G$. $N\cap M = \{e\}$. Prove that $nm = mn$ for every $n\in N,m\in M$. What i did - I know that $gng^{-1}\in$ N for ...
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1answer
47 views

Prove that all normal subgroup definitions are equivalent.

given $ N<G $ I need to prove that all of the below are equivalent: 1) for each $g \in G$ , $n \in N$ $gng^{-1} \in N $ 2) for each $g \in G$ $gNg^{-1} = N $ 3) for each $g \in G$ $gN = Ng$ ...
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3answers
75 views

Show that $D$ is a normal subgroup.

Let G be a group. Let $D$ be the subgroup of $G$ generated by the elements of the form $ghg^{-1}h^{-1}$, where $g,h\in G$. Show that $D$ is a normal subgroup. I am having trouble showing that $D$ ...
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4answers
47 views

Order and index of a normal subgroup $N$ are relatively prime

Let $N$ be a normal subgroup of a finite group $G$. Assume that the order of $N$ and the index of $N$ in $G$ are relatively prime. Prove that if $g\in G$ satisfies $o(g)\mid o(N)$, then $g\in N$. ...
2
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1answer
30 views

Disjoint normal subgroups - one contained in the centralizer of the other

Let N and M be normal subgroups of a group G and assume that N and M have only one element in common. Prove that N is contained in $C_G(M)$. First I concluded that |NM|=|N|*|M|. Now I'm trying to ...
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1answer
59 views

Is $O(n)$ normal in $GL(n)$?

Is the orthogonal group $O(n)$ normal in $GL(n)$? Here is what I did so far: Let $Q\in O(n),S\in GL(n)$ we want to check if $S^{-1}QS\in O(n)$: $(S^{-1}QS)^T=(S^{-1}QS)^{-1}\iff ...
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2answers
36 views

$H,N(H)$ are subgroups of $G$ show that $H\lhd N(H)$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ $N(H)$ is also subgroup of $G$. I need to prove that $H$ is a normal subrgoup in $N(H)$ Attempt: $H\lhd N(H) \iff ...
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1answer
29 views

prove that if $N\lhd G$, $ M\lhd G$, $M\bigcap N=\{e\}$ so: $mn=nm , \forall n \in N,\forall m\in M$ [duplicate]

prove that if $N\lhd G$, $ M\lhd G$, $M\bigcap N=\{e\}$ so: $mn=nm , \forall n \in N,\forall m\in M$
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1answer
21 views

The Image of normal subgroup is also normal subgroup?

Let $G$ be a group , $N\lhd G$ , $ \varphi:G\rightarrow G'$ is homomorphism onto $G'$, prove that $\varphi(N)=\{\varphi(n):n\in N\}$ is normal subgroup of $G'$
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1answer
50 views

For any finite group $G$ and for any natural number $n$, does there exist a group $H$ such that $|H|=n|G|$ and $G$ is a normal subgroup of $H$? [closed]

For any finite group $G$ and for any natural number $n$, does there exist a group $H$ such that $\left\vert H\right\vert=n\left\vert G\right\vert$ and G is a normal subgroup of H?
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1answer
38 views

Abstract Algebra: Proof involving a p-Sylow, nomal subgroup P in G. [closed]

Let $P$ be a normal subgroup in $G$ where $P$ is a $p$-Sylow subgroup of $G$. Show that $\phi(P)=P$ for every automorphism $\phi$ of $G$.
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1answer
18 views

If $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ show that $N\lhd G$

Let $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ I need to prove that $N\lhd G$ Attempt: $N\lhd G \iff gng^{-1}\in N$ and it is in $N$ for all $g\in G, n\in N$
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5answers
119 views

Prove every group of order less or equal to five is abelian [closed]

Is it possible to prove that every group of order less or equal to five is abelian? thanks,
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2answers
267 views

Every subgroup of a normal subgroup is normal

Is every subgroup of a normal subgroup normal ? That is if $H$ is a normal subgroup of a group $G$ and $K$ is a subgroup of $H$, then $K$ is a normal subgroup of $G$. Is it true ? If not what is the ...
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1answer
28 views

Proving Quotient group is cyclic / abelian via Isomorphism theorem

Let $G$ be a group with normal subgroup $N$, then $G/N$ is the quotient group. If $G$ is cyclic, let its generator be $a$, then $Na$ is the generator for $G/N$. [Since $Nx = Na^i = (Na)^i$.] If $G$ ...
0
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2answers
35 views

Some very short clarification on quotient groups

This is something I'm slightly confused with: I have a group, $S_3$, say, and its normal subgroups $N$ are $\{e\}, A_3$ and $S_3$. Then its quotient groups are thus $S_3/N$ for each $N$. But what ...
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2answers
80 views

Group of order $135$ abelian and not cyclic

I am trying to solve the following: Let $G$ be a group of order $135$. Show that if $G$ has more than one normal subgroup of order $3$, then $G$ is abelian and non-cyclic. What I could do was: ...
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0answers
25 views

What is the Schreier graph of this group/subgroup/generating set?

Let $G=\langle Pa, b\mid a^3 = \mathrm{id}, (ba)^2 = \mathrm{id}\rangle $, $H = \langle b\rangle$ , and $S = \{a, b\}$. I have been trying to figure out what elements are in this group by finding the ...
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1answer
22 views

Please help me with proving a subgroup is normal

Question: Let $G$ be a group and $N = \{g_1h_1g_1^{-1}h_1^{-1} \dots g_nh_ng_n^{-1}h_n^{-1} \mid n \in \Bbb N, g_i, h_i \in G\}$. Show that $N$ is a normal subgroup of $G$. I have shown that $N$ ...
1
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1answer
52 views

Normal subgroups and index problem

Let $G$ be a finite group and $H$ and $K$ subgroups such that $H \lhd G$ or $K \lhd G$. If $gcd(|K|;|G:H|)=1$, show that $K \subset H$. I think I could prove it for the case $H \lhd G$ Let $k \in ...
3
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1answer
16 views

No normal subgroup of a subgroup of $S_n$ imply the subgroup is the one of even permutations or consists of two elements

The following is an old exam question from a n introduction to group theory course: Let $G$ be a proper subgroup of $S_{n}$, $n\geq3$. Prove that if $G$ does not have any non-trivial normal ...
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1answer
28 views

If $G$ acts non-trivially on a set then it has a proper normal subgroup of finite index

The following is an old exam question from a n introduction to group theory course: Let $G$ be a group (possibly infinite) that acts non-trivially on a finite set $X$ - that is there are $g\in ...
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1answer
34 views

Is the intersection between a subgroup, and a normal subgroup, normal in the parent group?

Given $H, N \subset G$ and $N \lhd G$ is there some underlying fact or theorem for why $H \cap N$ would or would not be normal in $G$? My reasoning would be that it would be normal to $G$ as; $\forall ...
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2answers
32 views

Show H is normal to G iff $N_G(H)=G$

Let $G$ be a group and $H\leq G$. The normalizer of $H$ in $G$ is $N_G(H)=\{ g\in G |gHg^{-1} =H \}$. Show H is normal to G iff $N_G(H)=G$ I know that $H$ is normal to $N_G (H)$, for the inverse ...
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1answer
18 views

Show that $N(H)$ is the largest subgroup of $G$ in which $ H$ is normal.

Let $G$ be a group and $H \leq G$. The normalizer of $H$ in $G$ is $$N(H) = \{g \in G|gHg^{−1} = H\}$$ If $H$ is a normal subgroup of $K \leq G$ then $K \leq N(H)$. Show that $N(H)$ is the largest ...
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0answers
19 views

Example of not normal subgroup $N$ such that not all orbits of $X$ under $N$ have the same length

If $G$ is a finite group that works transitively on $X$, what is a simple example of a subgroup $N$ that is not normal such that not all orbits of $X$ under $N$ have the same cardinality?
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2answers
48 views

Action of $G$ on the left cosets of $H$ giving a non-trivial homomorphism

If $H < A_5$ is a subgroup of index $3$, the action of $G$ on the left cosets of $H$ gives a non-trivial homomorphism $$\underset{order \ 60}G \rightarrow \underset{order \ 6}{S_3}$$ which ...
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0answers
60 views

Wanted: Group homomorphism $\operatorname{Aut}(G) \to H$ whose kernel is the set of inner automorphisms

By a straightforward computation, it is not hard to show that the set $\operatorname{Inn}(G)$ of the inner automorphisms of a group $G$ is a normal subgroup of $\operatorname{Aut}(G)$, see for example ...
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2answers
50 views

A question about direct product of subgroups.

Let $H$ and $K$ be two subgroups of a group $G$. Suppose that $H$ is normal in $G$ and $G/H\simeq K$. My question is when $G\simeq H\times K$? My guess is if $K$ is normal in $G$ and $G=HK$ then ...
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2answers
36 views

Do groups of prime order have one subgroup?

So let's say that I have a group of order $p$, where p is prime; does that group only have one subgroup? I've look at the wiki article and it says there's a trivial and actual solution, so can we ...
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1answer
32 views

Are these example subgroups?

Example 1 The subgroup $\{0,3\}$ of $\mathbb Z_6$ is a normal subgroup I know it's a subgroup because the element $3\in\mathbb Z_6$ generates $\{0,3\}$ under addition $\langle 3\rangle=(0,3)$ and ...
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0answers
90 views

Is $A_4$ isomorphic to $D_3\times \mathbb Z_2$?

Prove/Disprove that $A_4\cong D_3×\mathbb Z_2$: I can't seem to find a way to disprove it (Something in my mind it telling me it can't be true). But the order of both is 12, both are non cyclic, non ...
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1answer
23 views

If $G$ is a finite group such that the Klein four group is a quotient of $G$ , then can we write $G$ as a union of three proper subgroups ?

If $G$ is a finite group such that their is a normal subgroup $H$ of $G$ such that $G/H \cong V_4$ , where $V_4$ is the Klein four group , then is it true that $G$ can be written as a union of three ...
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3answers
58 views

Case when all subgroups of a group are normal

I was thinking in light of this question, where the groups G in discussion is of order 21 and question is to find number of non-trivial normal sub-groups of G. In the answers it is mentioned that ...
2
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0answers
59 views

Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

I have found all the subgroups of the Galois group of the splitting field $K=\mathbb{Q}(\sqrt[8]{2},i)$ over $\mathbb{Q}$, and I know that if any of the subgroups $H$ of $G=$Gal$(K/\mathbb{Q})$ are ...
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3answers
76 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
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1answer
85 views

The Union of Two Normal Subgroups is also a normal subgroup

I know the statement The Union of Two Normal Subgroups is also a normal subgroup is false. Is there a counter example to show this? I can prove that the intersection is normal, but I can't disprove ...
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2answers
80 views

When is a group isomorphic to a proper subgroup of itself?

A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?
1
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1answer
22 views

Consider $\phi: G \to S_4\times D_{15}$ a homomorphism and onto

Q1) Prove that $G$ contains an element of order 20 Q2) Assume $\exists H\subset G$ s.t $H$ normal in $G$ and |$\phi(H)$|=60. Prove that $G$ contains a normal subgroup $K$ such that |$G/K$|=36 For ...
3
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1answer
83 views

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$.

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$. We can suppose $P \subseteq K\lhd G$. Since $P$ is ...
1
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1answer
77 views

$G$ is a finite group,with eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ s.t. index $[G:N]$ divisible by 56, not by 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. I will start by ...
4
votes
2answers
53 views

If $K$ and $H$ are subgroups of $G$ and $H \triangleleft K$ then $K \subseteq N(H)$.

We can easily prove the truth of this statement. My question is that why do we not simply say $K=N(H)$ $?$ I'd be really grateful for an elaboration on this.
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0answers
41 views

Problem about finite group whose proper subgroups are abelian. [duplicate]

Let $G$ be a finite group with the property that all of its proper subgroups are abelian. Let $N$ be a proper normal subgroup of $G$. Show that either $N$ is contained in the center of $G$ or else $G$ ...