Questions concerning normal-subgroups of groups.

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2answers
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Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H. [on hold]

Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H. Show that if K is a normal subgroup of G, then K is a normal subgroup of H.
3
votes
1answer
50 views

Which non-abelian finite groups have the property that every subgroup is normal?

If $G$ is an abelian group, every subgroup $H$ of $G$ is normal. I searched for non-abelian finite groups $G$ , such that every subgroup is normal and GAP showed only the groups $G'\times Q_8$ , ...
0
votes
1answer
29 views

Normal Subgroups and Properties

Suppose we have the normal subgroups $H,J\subset G $ with the property $|G|=|H|\cdot |J|$ and $H\cap J={e}. $ $ $Prove that $H\times J\cong G$. I don't really know how to approach this one. I ...
1
vote
2answers
34 views

$G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$?

Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing ...
0
votes
1answer
65 views

Intersection of two normal subgroups of a group

Let G be a group, and let A,B be normal subgroups of G. If $a \in A$ and $b \in B$, does this mean that $ab \in A \cap B$?
0
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0answers
8 views

Relation between $\prod^t_i S_{N_i} \wr D_{m_i} $ and $A_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of the symmetric group $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ ...
1
vote
1answer
36 views

Number of normal subgroups from $F_{2}$ which factor groups are isomorphic to $Q_{8}$

What is the number of normal subgroups of the free group of rank 2 $F_{2}$ whose factor groups are isomorphic to the $Q_{8}$? I have a plan solutions. But I can't get a numerical answer. Minimum ...
1
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0answers
8 views

Can there be a tower like $1 = A_0 \triangleleft \ldots \prod^t_i S_{N_i} \wr D_{m_i} \ldots \triangleleft = S_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral ...
1
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0answers
30 views

Easy to generate subgroups of the symmetric group $S_n$

Which subgroups of the symmetric group $S_n$ can be generated in polynomial or subexponential time?
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0answers
14 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a maximal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ is a maximal subgroup of $S_{\sum^t_i N_i m_i}$? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ ...
5
votes
1answer
43 views

If $G$ is a direct product of simple groups, then is every simple subgroup of $G$ isomorphic to a subgroup of some factor?

Let $G=N_1\times N_2\dots \times N_n$. Suppose that $H$ is a simple subgroup of $G$. Is $H$ isomorphic to a subgroup of $N_i$, for some $N_i$? This is a weaker version of this question, which turned ...
0
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1answer
28 views

$H,K$ normal subgroups of a finite group $G$ , $G \cong H \times K$ , every element of $H$ commutes with every element of $K$ , then is $G=HK$?

Let $H,K$ be normal subgroups of a finite group $G$ such that $G$ is isomorphic with $H \times K$ and every element of $H$ commutes with every element of $K$ , then is it necessary that $G=HK$ ? ( ...
0
votes
1answer
32 views

$H,K$ be normal subgroups of $G$ such that $G$ is isomorphic with $H \times K$ , then is $G=HK$?

Let $H,K$ be normal subgroups of $G$ such that $G$ is isomorphic with $H \times K$ , then is it necessary that $G=HK$ ? What if we also assume that every element of $H$ commutes with every element of ...
2
votes
2answers
50 views

$H,K$ be subgroups of a group $G$ such that $G$ is isomorphic with $H \times K$ ; then is $H$ normal in $G$ ?

Let $H,K$ be subgroups of a group $G$ such that $G$ is isomorphic with $H \times K$ ; then is $H$ normal in $G$ ? I can prove that $H \times \{e\}$ is normal in $H \times K$ but nothing else ; one ...
1
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1answer
26 views

Prove that $\langle S \rangle = \bigcap_{S\subseteq H, H\le G}H$

Prove that $\langle S \rangle = \bigcap_{S\subseteq H, H\le G}H$ where $\langle S \rangle = \{x_1, . . . , x_m \mid x_i \in S\cup S^-, m \in \mathbb{N} \}$ I know how to prove that $\langle S ...
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0answers
16 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ ...
4
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1answer
57 views

Every group of order $60$ , having a normal subgroup of order $2$ , has a normal subgroup of order $6$ (without Sylow )?

How to prove , without using Sylow's theorems , that every group of order $60$ , having a normal subgroup of order $2$ , contains a normal subgroup of order $6$ ? Please help . Thanks in advance
1
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1answer
19 views

If the order and index of a normal subgroup are relatively prime, then it is a characteristic subgroup

If $H$ normal in $G$ and $(|H|,[G:H])=1$ then $H$ char $G$. Honestly, I haven't the least idea how this can be proved, and I have tried a lot. Any hints?
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1answer
19 views

Prove that $AN$ is a subgroup of $G$ if $A$ and $N$ are its subgroups and $N$ is normal in $G$

Let $A$ be a subgroup of a group $G$, and let $N$ be a normal subgroup of $G$ Prove than in this case $AN = \{an|a \in A, n \in N \}$ is a subgroup of $G$ I know that since $N$ is normal, we have $AN ...
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1answer
28 views

Normal Subgroup and Equal Cosets Proof

I started learning group theory and got stuck with one proof... A normal subgroup $N \subseteq G$ is defined as: $\forall g \in G; n \in N \space\ (g*n*g^{-1} \in N) $ How can I show that, using ...
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2answers
42 views

Every subgroup of a quotient group is a quotient group itself

Let $G$ be a group and $N$ its normal subgroup. Now, let $B$ be a subgroup of $G/N$. I need to prove that $B = A/N$ for some subgroup $A$ of $G$ that contains $N$. Here's what I did: Given a normal ...
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1answer
24 views

$N$ is a normal subgroup of $G$, when does $o(Na)=o(a)$

If $N$ is a normal subgroup of $G$, I know that $o(Na)|o(a)$ for $a \in G/N$. But in what cases does $o(Na)=o(a) $ ?
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1answer
24 views

Showing normal subgroup is a subgroup of the center.

Given $N\lhd G$ and $N \cap G'=e$. Show that $N\leq C(G)$ where $G'$ is commutator group. What exactly do we get from the hypothesis $N \cap G'=e$?
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1answer
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Problem with Cosets in the proving of $K/H \trianglelefteq G/H$

We have $H \trianglelefteq G$ and $H \subseteq K \leq G$. we can prove if $K \trianglelefteq G$ then $K/H \trianglelefteq G/H$ like this: We already showed that $K/H \leq G/H$. If we show that ...
2
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0answers
34 views

How to show that $(H,\cdot)$ is normal subgroup of $GL_2(\Bbb R)$?

Let $H=\left\{\left| \begin{matrix} a&b \\0&d\end{matrix} \right| :a,b,d\in \Bbb R \text{ and } ad\neq 0\right\}$. Show that $(H,\cdot)$ is a normal subgroup of $GL_2(\Bbb R)$ . is this a ...
1
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1answer
65 views

When is a group isomorphic to the product of normal subgroup and quotient group? [closed]

Let $H$ be a normal subgroup in $G$. When is $G$ isomorphic to $H\times (G/H)$? I think it's always true in the abelian case. Are there other rules?
0
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1answer
55 views

Normal subgroup is permutable!

Where can I find the proof of : The normal subgroups are always permutable?? The group is permutable if $HX=XH$ for every subgroup $X$ of group $G$( $H$ is a subgroup of the group $G$).
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4answers
49 views

Subgroup with index equal to smallest prime factor normal. How can I prove this?

Let $G$ be a group of order $n>1$ and $p$ the smallest prime factor of $n$. Suppose, $H$ is a subgroup of $G$ and $[G:H]=p$. How can I prove that $H$ is normal ? According to Lagrange, ...
0
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1answer
13 views

Normal subgroup of Engel group

The Engel algebra $\mathfrak g$ is the Lie algebra generated, as a vector space, by four vectors $X_1,X_2,X_3,X_4$ with the only non trivial commutation relations:$$[X_1,X_2]=X_3, \quad ...
6
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1answer
75 views

Group Theory : What is $Ha \ne Hb$?

As a beginner of Group Theory, I got stuck with the following question: Suppose that $H$ is a subgroup of $G$ such that whenever $Ha \ne Hb \space ,$ then $aH \ne bH$. $(a,b \in G)$ Prove that ...
1
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3answers
60 views

If $ A \trianglelefteq G $, and $ B \trianglelefteq A $ is a Sylow subgroup of $ A $, then is $ B \trianglelefteq G $?

Let $ A \trianglelefteq G $ and $ B \trianglelefteq A $ a Sylow normal subgroup of $ A $. My textbook says then that $ B \trianglelefteq G $. I don’t understand why that is.
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1answer
60 views

Showing that every subgroup of an abelian group is normal [duplicate]

I'm working on a proof to show that every subgroup of an abelian group is also a normal subgroup. Let $G$ be an abelian group and $H$ an arbitrary subgroup of $G$. I want to show that $gHg^{-1} = H$, ...
0
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1answer
66 views

Some questions about (normal) subgroups of finite groups

If the number of subgroups $H$ of some finite group $G$ is mentioned, as far as I understood, all subgroups are considered, no matter whether two or more of them are isomorphic, whereas the number of ...
0
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2answers
16 views

How can I prove that the order of the group $G$ is equal to the order of each coset multiplied by the number of cosets?

Suppose you have a group $G$ with some subgroup $H$. How can I show that the order of $G$ is equal to the order of each coset of $H$ in $G$ multiplied by the number of different cosets? Thanks
0
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1answer
28 views

Let $G,H$ be two groups. Show that the order of $G$, $|G| = 16$. [closed]

Let $G = U(\mathbb{Z}_{32})$ and let $H = \big\{[1],[31]\big\} \leq G$. (i) Show that $|G| = 16$.
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0answers
9 views

Define a function map

Given $\sigma : G → (G/M) $ x $G/N$. Define $\sigma$. Can someone please check if I have defined it well. $\sigma(g) = ( g + M, g+ N)$ or $\sigma(g) = ( gM, gN)$ ? would either work given $M,N$ ...
4
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1answer
38 views

Find a subgroup of $GL_3(\mathbb{Z}_8)$ of index 2

On my final exam yesterday there was one "almost bonus" question which I don't really think I did right. I "guessed" that such a subgroup is $GL_3(\mathbb{Z}_4)$. A hint would be appreciated. The ...
0
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2answers
58 views

Show that $\langle S\rangle$ is a normal subgroup of $G$.

Let $S$ is a subset of a group $G$, and $gsg^{-1} \in S$ for each $s \in S$ and $g \in G$. Show that $\langle S\rangle$ is a normal subgroup of $G$. I am stuck on this problem. Is anyone is able to ...
1
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1answer
28 views

Proving $A_5$ is the smallest perfect group.

I am trying to find a streamlined proof that $A_5$ is the smallest perfect group, with respect to order ( a group is perfect if it has no abelian quotients except $\{e\}$). I know it follows directly ...
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1answer
18 views

$\overline{N_P(H)} = N_\overline{P}(\overline{H})$?

Let $P$ be a $p$-group. For a subgroup $K$ of P containing $Z(P)$, by $\overline{K}$ we denote the quotient $K/Z(P)$. Let $H$ be a proper subgroup of $P$ containing $Z(P)$. The proof of the first ...
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2answers
25 views

Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

I am trying to find Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$. For instance, The subset $\mathrm{T}(n,\mathbb{R})\subset \mathrm{GL}(n, \mathbb{R})$ of ...
4
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2answers
56 views

Does there exist a subgroup $G$ of $\mathbb{R}$ such that $\mathbb{R}/G \cong \mathbb{Z}$?

My question is whether or not it's possible to find an additive subgroup $G$ of the real numbers such that the quotient group $\mathbb{R}/G$ is isomorphic to the infinite cyclic group. I'm not ...
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0answers
39 views

If $H\leq G$ of index $p$ in $G$ and $p\mid |G|$ then $H\trianglelefteq G$ [duplicate]

Given that $H\leq G$ of index $p$ in $G$, where $p$ is the smallest prime integer such that $p \mid |G|$, then $H \trianglelefteq G$. I would appreciate some hints, as I don't even know where to ...
1
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3answers
85 views

Is this true for any subgroup of a group?

If $H\leq G$, then is it true that $g_1 g_2 H=g_1 H g_2 H$ for any subgroup $H$, or only if $H$ is normal in $G$? And how is this equality proved? Otherwise, if the equality above is true for all ...
1
vote
1answer
93 views

Finding the center and normal subgroup of $D_{10}$

Consider $D_{10}$, the dihedral group of order $10$. This is the group of symmetries of the regular pentagon. Label the vertices of the pentagon clockwise as $1,2,3,4,5$. Let $\sigma = (12345)$ and ...
0
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1answer
18 views

Quotient group of invertible matrices isomorphic to $\mathbb{R}\times$

I need to show that the quotient group $G/H$ is isomorphic to $\mathbb{R}×$; where $G = \{ A = (a b 0 d): \det A \neq 0 \}$, $H = \{B = (a b 0 a): \det B \neq 0\}$, $\mathbb{R}\times$ is the group of ...
4
votes
3answers
85 views

Determine if H is a normal subgroup of G - faster way than finding cosets?

$G = S_4$, $H = \{(1),(12)(34),(13)(24),(14)(23)\}$. I just did it the long way, and found H to be normal. Is there a better way than finding left and right cosets? I don't want to spend this kind ...
1
vote
0answers
55 views

Which lines in $\mathbb R^2$ define a subgroup?

Which lines in $\mathbb R^2$ define a subgroup? I know that the line $y=x$ in $\mathbb{R}^2$ gives a subgroup, but I can't figure out the other ones.
0
votes
1answer
40 views

Having Trouble with Groups

Suppose that $H$ and $K$ are groups, $G = H \times K$. $A = \{(h,1): h \in H\}$. (a) Prove that $A$ is a normal subgroup of $G$ isomorphic to $H$. (b) Prove that the factor group $\frac GA$ is ...
0
votes
2answers
55 views

Show the normal subgroups and cosets of a dihedral group (D6)

$G=D_6$ and $H=<R^2>$. Use this Cayley table for $D_6$ (a). Show that $H \vartriangleleft G$. I want to show by finding out $aH=Ha$ for all $a \in G$, but then how do I proceed, it would be ...