Questions concerning normal-subgroups of groups.

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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 ...
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1answer
30 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
64 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
18 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
42 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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34 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
68 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
39 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
54 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
75 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 ...
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1answer
67 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
51 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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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
124 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
45 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 ...
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1answer
38 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 ...
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2answers
78 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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22 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 ...
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1answer
56 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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2answers
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Show that $A_3$ is a normal subgroup of $S_3$. [closed]

Show that $A_3$ is a normal subgroup of $S_3$. Is it possible to prove this without separately showing $gHg^{-1} \in H$ for every element?
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1answer
43 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?
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35 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 ...
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1answer
41 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 ...
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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 $D_{n}$?
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Is it a coincidence that $\mathbb{Z_{12}}$ has all distinct subgroups that are factors of 12?

So I'm doing a problem where I need to find all the distinct subgroups of $\mathbb{Z_{12}}$ and I found them to be <1>, <2>, <3>, <4>, <6>, and <12>. They're all factors of 12 and ...
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2answers
55 views

Problem about the subgroup generated by the commutators.

I want to make a question about this exercise. Let $G$ be a group, and let $a$, $b$, be elements of $G$. We define the commutator of $a$ and $b$ as follows: \begin{equation} ...
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47 views

Let $H,K$ be finite groups , if for any finite group $G$ , $h(G,K)=h(G,H)$ holds , then is it true that $i(G,H)=i(G,K)$ for any finite group $G$ ?

Let $G,H$ be two groups , let $h(G,H)$ be the number of all group homomorphisms from $G$ to $H$ and $i(G,H)$ be the no. of all injective group homomorphisms from $G$ to $H$ . I know the relation ...
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1answer
48 views

If $H$ is a normal subgroup of $G$, prove $g^{n} \in H$ [duplicate]

I'm having trouble with proving the following: Let $H \subset G$ be a subgroup with finite index $n = [G:H]$ Prove: $H$ is a normal subgroup of $G$ $\Rightarrow g^{n} \in H$ $\forall$ $g \in G$ So ...
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34 views

Quotient group with multiplication table involving the Klein 4 subgroup.

Show that the subgroup $V=\{e, (12)(34), (13)(24), (14)(23)\}$ is normal in $S_4$. Make a multiplication table for the quotient group of $S_4$ and $V$ Proving that $V$ is a normal subgroup is not a ...
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3answers
34 views

Size of the product of two subgroups

Let $(G, \ast)$ be a group and let $H\le G$ and $K\le G$ be subgroups of $G$. Prove that $|HK|$=$\frac{|H|\cdot|K|}{|H\cap K|}$. Intuitively this is quite obviously true, as otherwise the products of ...
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1answer
21 views

Coset to a power

my book states that if $N \trianglelefteq G$ then $(gN)^\alpha = g^\alpha N$ for $\alpha \in \mathbb{Z}$ The proof should be simple by induction but I can't understand how $(gN)^0 = [N = g^0N]$. How ...
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1answer
36 views

Group and subgroup generated by element in G and application of Lagrange' s theorem

It might be a stupid question. Let G be a group and H be a subgroup If there is "a" in H, then subgroup generated by a is a subgroup of H Then, order of subgroup generated by a must divide the ...
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3answers
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Prove: $G/(N_1 \cap N_2)$ is isomorphic with a subgroup of $(G/N_1) \times (G/N_2)$

I have to solve this exercise for my math study, but don't know how to do it. It's keeping me busy for 2 days now. Let $G$ be a group and $N_{1}, N_{2}$ normal subgroups of $G$. Let $f: G \rightarrow ...
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0answers
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Why is it necessary for a quotient group to be defined for a normal *subgroup*.

So I was thinking about the Second Isomorphism theorem, and noticed that quotient group $(A + B)/B$ could be written more nicely as $A/B$ if it were not required for the group representing the ...
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2answers
33 views

Suppose that $G$ is a group and $A \lhd G$. Prove that if $A$ is abelian and $G=AH$ for some subgroup $H$ of $G$, then $A \cap H \lhd G$

Suppose that $G$ is a group and $A \lhd G$. Prove that if $A$ is abelian and $G=AH$ for some subgroup $H$ of $G$, then $A \cap H \lhd G$ So, I need to prove that $H \cap A \lhd G$ How do I go about ...
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53 views

Commutativity between elements belonging to two normal subgroups of a given group G.

Let $G$ be a group. Given two normal subgroups $N$ and $M$ in $G$ whose intersection is the identity element, show that for every $n,m$ belonging to $N$ and $M$ respectively, $nm=mn$.
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Let $K<G$, where G is a group. Define $N_G(K)=(g\in G/gkg^{-1}=K)$ Prove the following:

Let $K<G$, where G is a group. Define $$N_G(K)= \{g\in G \mid g K g^{-1}=K\}$$ which is called the normalizer of subgroup K. 1.) Prove that $N_G(K)$ is a subgroup of G and K is a normal subgroup ...
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1answer
40 views

Can a normal subgroup of a finite nonabelian group be nonabelian?

We know that if a group is Abelian, then all its subgroups are normal. Also, if a group is nonabelian, it can contain a subgroup which is Abelian. Eg: The Dihedral group of order 2n, $D_{2n}$ is ...
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1answer
21 views

Why is a semigroup $H$ of prime size with proper subgroup $G$ not group?

I know that this is addressed in a corollary of Langrange's theorem, which states that if a group $H$ has a prime order, there exists no nontrivial subgroups, since the cosets of any subgroup must ...
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40 views

Classifying groups of order $8$

Given group $T$ of order $8$, and $t \in T$ such that $ord(t) = 4$. Let $P = \{1,t,t^2,t^3 \}$ and let $x \in T−P$. List possibilities for $x^2$ labelling as $(a_1,a_2, \ldots ,a_n)$. List ...
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125 views

If we have a subgroup of index 3 that is not normal, show there is a subgroup with index 2

Given a a subgroup $H$ of $G$ with index $3$, we have to show there is a subgroup $K$ of $G$ with index $2$, assuming that $H$ is not a normal subgroup of $G$. My line of thinking was the ...
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1answer
43 views

The intersection of subgroup and normal subgroup: the greatest common divisor?

Is the order of intersection of subgroup $H$ and normal subgroup $N$ of group $G$ the greatest common divisor of $\lvert H\rvert$ and $\lvert N\rvert$?
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2answers
31 views

Prove: $aH = bH \iff Ha^{-1} = Hb^{-1}$

I have to prove this exercise for my math-study: Let $G$ be a group and $H \subset G$ a subgroup. Prove that for every $a,b \in G$ holds: $$aH = bH \iff Ha^{-1} = Hb^{-1}$$ I tried this, but I'm ...
3
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3answers
56 views

Let $G$ be a group and $H,K,L$ be subgroups with $H \subseteq K$. Prove that $K \cap HL=H(K\cap L)$

Let $G$ be a group and $H,K,L$ be subgroups with $H \subseteq K$. Prove that $K \cap HL=H(K\cap L)$ Ok, so I know that $H,K,L$ are subgroups of $G$ which tells me that $H,K,L$ are groups themselves ...
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1answer
36 views

If [H,N]≠1 then H∩N≠1 and N≤H . Now NOp′(H) ,so that N∩Op′(H)=1. Hence N is a p-group. Now can we say CH(N)≥Op′(H) and H/CH(N)is p-group?

If $G$ is a group. Let $H$ be a normal $p$-nilpotent subgroup of $G$ and let $N$ be a minimal normal subgroup of $G$ whose order is divisible by $p$. If $[H,N]\neq 1$ then $H \cap N \neq 1$ and $ N ...