Questions concerning normal-subgroups of groups.

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1answer
29 views

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

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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0answers
13 views

does dicyclic group of order $2^{\alpha}m$, $(2,m)=1$ have normal subgroup of order $m$?

consider dicyclic group with presentation $$Dic_n=< a,b |a^{2n}=1,a^n=b^2,ba=a^{-1}b>$$ the order of this group is $4n$,suppose that $|Dic_n|=2^{\alpha}m$ where $(2,m)=1$ my question is about ...
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1answer
13 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
89 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
55 views

Prove $Inn(G)$ is a normal subgroup of $Aut(G)$ [closed]

I found documents explaining how to prove $Inn(G)$ is a normal subgroup of $Aut(G)$, but can we prove for $G$? Also, I have another question along the same line. How do we show $Aut(G)/Inn(G) \cong ...
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0answers
53 views

The Existance Of Isolated Subgroups & How can you adjacency-listrially represent the above? [closed]

In a group which has 'N' members, What is the number of cases that all the members are connected with one another, not being isolated to sub-groups?       N=1 1   N=2 1   ...
3
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2answers
243 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
21 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 $G$ ...
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2answers
31 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
44 views

Subgroups/Normal [closed]

Given that $G=(S_3,\circ)$ and $H=\langle(2~1~3)\rangle$. I have to show that $H$ is not a normal subgroup of $G$. I am a little lost outside of showing that right and left cosets don't match.
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2answers
73 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
19 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 ...
2
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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
51 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
26 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
33 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
30 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
14 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
36 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
52 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
49 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
35 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 ...
0
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1answer
31 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
88 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
22 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
50 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 ...
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0answers
50 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 ...
2
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3answers
73 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
58 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
61 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?
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1answer
20 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 ...
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1answer
80 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
71 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 ...
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2answers
52 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 $G$ ...
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1answer
125 views

If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. Now this is my ...
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1answer
30 views

Prove that $H$ is a normal subgroup of $G$. [$aH =Hb$]

Let $H$ be a subgroup of a group $G$. If for each $a \in G$, there exists $b \in G$ such that $aH = Hb$, then prove that $H$ is a normal subgroup of $G$. How do I proceed?
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1answer
46 views

Find the order of $Z(G)$.

If $G$ be a group of order $pq$ where $p$ and $q$ are prime integers then find $|Z(G)|$. The options are i) 1 or $p$ ii) 1 or $q$ iii) 1 or $pq$ iv) None of these. I know groups of prime order are ...
1
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1answer
43 views

Prove that if H and K are normal nilpotent subgroups of a finite group G, then HK is a normal nilpotent subgroup.

HK us the group composed of the products of the elements in H and K under multiplication in G, but it is not the direct product (that would be so much easier). I have figured out the normal subgroup ...
3
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2answers
80 views

Is it true that $aH = bH$ iff $ab^{-1} \in H$

Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$. My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? ...
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0answers
23 views

Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
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1answer
36 views

Prove: $G \cong M \times N$ and $G$ is finite $\Rightarrow order(N)$ is not divisible by 5

I have the following questions about things I have to prove for my math study: Let $G$ be a group with the property that $(gh)^5 = g^5h^5$ for all $g, h \in G$. Let $M = $ {$g \in G | g^5 = e$} and ...
1
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1answer
58 views

Show that H is normal subgroup of G. [duplicate]

Let H be a proper subgroup of G and a $\in$ G, a $\notin$ H. Suppose that for all b $\in$ G, either b $\in$ H, or Ha = Hb. Show that H is normal subgroup of G. How do I proceed on this?
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1answer
44 views

Find the number of normal subgroups of $A_4$.

Find the number of normal subgroups of $A_4$. Is there any way to find this without actually finding the subgroups?
0
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2answers
39 views

Prove H is a normal subgroup of G.

Let $G$ be a group and $H$ a subgroup of $G$. If for all $a, b \in G, ab \in H$ implies $ba \in H$, then prove that $H$ is a normal subgroup of $G$. How do I proceed on this? I tried to prove for all ...
3
votes
1answer
44 views

Prove: $G = HN$, and $H \cap N =$ {$e$}, (Isometries in $\mathbb{R^2}$)

I'm trying to solve this problem: Let $G = E(\mathbb{R^2})$ be the group of all isometries of $\mathbb{R^2}$, so $G$ consists of translations, rotations about the origin and reflections in a line ...
1
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1answer
59 views

proof: every $g \in G$ can be written as $g =nm$

Let $G$ be a group, and $N$ and $M$ normal subgroups of $G$. Also $N \cap M =$ {$e$} Why is it true that if G is generated by $N \cup M \Rightarrow$ every $g \in G$ can be written as $g = nm$ with $n ...
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2answers
59 views

Number of normal subgroups from $F_{2}$ which factor groups are isomorphic to $D_{n}$

What is the number of normal subgroups of the free group $F_{2}$ whose factor groups are isomorphic to the dihedral group $D_{n}$?