Questions concerning normal-subgroups of groups.

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4answers
46 views

Prove that the 4-group V is normal subgroup of $S_4$ by using isomorphism theorem

Prove that the 4-group V is normal subgroup of $S_4$ First, by using the multiplication table, I am able to prove that 4-group V is subgroup of $S_4$. But I face problem in proving that $\forall ...
2
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2answers
66 views

Does $AN/N=A/N$?

I had a debate with my professor about this today because he assigned the following problem. It is all very simple, yet it seems to me that the results are contradictory. Let $G$ be a group and ...
2
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1answer
27 views

Homomorphic image if smaller fails to exist

Suppose a finite group $G$ has no homomorphic image of order $n$. Is it possible for $G$ to have a homomorphic image of order a multiple of $n$? My gut says "no", as the larger homomorphic image ...
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2answers
112 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
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3answers
123 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
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2answers
29 views

$H\subseteq G$, $N\triangleleft G$, and showing $|[G:N]|$ is a prime number

Let $G$ be a finite group and let $N\triangleleft G$ a normal subgroup. It is given that if $H\subseteq G$ such that $N\subseteq H\subseteq G$, then $H=N$ or $H=G$. Show that $|[G:N]|$ is a prime ...
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1answer
41 views

show that $\varphi : H \to H /N$ is one-to-one correspondence using kernel

I have a group G and subgroup N that is normal to G. As a part of proving the 3rd isomorphism theorm (a version of it) I need to prove that the transformation from the subgroups of G containing N ...
0
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1answer
32 views

Quotient group of free groups

Let $G=\langle g_1,\ldots,g_k\rangle$ be a free abelian group generated with $g_1,\ldots,g_k$ and let $H=\langle g_{r+1},\ldots,g_k\rangle$ be a free abelian subgroup of $G$. Is it then the case that ...
2
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1answer
54 views

Confusion with Centers, Conjugacy Classes, and Normal Subgroups

Ok this is a bit of a soft question, but I am in my first semester of algebra and getting continually confused by (what seems to me) some competing sets. Let $G$ be a group The center of $G$ is ...
2
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2answers
47 views

Normality of a normal subgroup of normal subgroup of G

Let $G$ be a non-Abelian Group and $H$ is normal subgroup of $G$. Is it always true that a normal subgroup $K$ of $H$ is also normal in $G$? Justify your answer. My answer is that, this is not true ...
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1answer
46 views

group $G$ of order $312$. show that G is not simple

I have a group G of order $312$ and I need to show $G$ is not simple. What I tried : I know $312 = 2^3\times39$ so, I know that I have an element of order $2$. does that mean I have a subgroup ...
3
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1answer
106 views

A question on subgroups of a finite group

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$. Prove that $K$ is a subgroup of $H$. So far we found that $o(K)$ divides ...
2
votes
1answer
29 views

Subgroup Lattice of Cosets

Use the subgroup lattice structure of the symmetries of a square, $Sq$, and use it to find the subgroup lattice of all left cosets of the normal subgroup $\langle r^2\rangle$. This is a problem ...
3
votes
1answer
48 views

Why is the commutator group a subgroup?

I am in Intro to Algebra, and have a question regarding the commutator subgroup. I am a bit confused with the premise, though, with how the set is a subgroup in the first place. Let $C$ be the set of ...
1
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1answer
52 views

Normal Submagma?

Is there a definition of normal submagma? visit https://en.wikipedia.org/wiki/Magma_(algebra) For normal sub-quasi-group I found two: A sub-quasi-group $H$ is called normal if there exists a normal ...
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2answers
28 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?) ...
0
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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 ...
0
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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 ...
3
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2answers
46 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
32 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
votes
3answers
74 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
42 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 ...
2
votes
1answer
54 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 $g \in G$ $gN = Ng$ ...
2
votes
3answers
77 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
52 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 ...
2
votes
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
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
147 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
297 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 ...
0
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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 ...
4
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2answers
86 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
31 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
54 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
17 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
29 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
33 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
50 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 ...
3
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0answers
65 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 ...