Questions concerning normal-subgroups of groups.

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18 views

show that for two normal subgroups with trivial intersection $n_1n_2 = n_2n_1$ [on hold]

Let $G$ be a group, and $N_1, N_2$ normal subgroups of $G$ s.t. $N_1\cap N_2 = \{e\}$. Show that $\forall n_1 \in N_1, \forall n_2 \in N_2, n_1n_2 = n_2n_1$.
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1answer
29 views

what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

I am looking for smallest example of a group $G$ such that: $G$ is a finite, non-abelian group $G$ is not simple $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $ $H_1, H_2, \dots $ ...
0
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1answer
25 views

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
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2answers
32 views

For all $g\in G$, is it true that $gHg^{-1} \subseteq H $ if H is a normal subgroup of G.

Assume that $H$ is a normal subgroup of group $G$. Is it true that for all $g∈G$ one has $gHg^{-1} \subseteq H$ ? I know that if H is a normal subgroup of G, then $ghg^{-1}$∈ $H$ where $h$∈$H$ and ...
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2answers
39 views

H prime order, normal subgroup of group G. Prove H in center Z(G).

I am looking at the following question: "Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime that divides the order of G. Prove that H is in the ...
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1answer
32 views

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$ [closed]

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$. Please, I appreciate any help, since I have some ideas, but those are ...
1
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1answer
66 views

Hatcher's proof of the van Kampen Theorem (injectivity of $\Phi$ – unique factorizations of $[f]$)

I am trying to understand the details of Allen Hatcher's proof of the Seifert–van Kampen theorem (page 44-6 of Algebraic Topology). My question is regarding the same part of the proof mentioned in ...
2
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2answers
72 views

The commutator subgroup $G'$ of $G$, is the set of all “long commutators”

By definition, the commutator is the subgroup generated by all commutators, that is $$G' = \hspace{1mm}<\{aba^{-1}b^{-1}|a,b \in G\}>$$ I'd like to prove that also $$X = \{a_1a_2\cdots ...
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votes
3answers
29 views

Normality is not transitive

Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p= \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix} $, let $L=(p)$, $K=L\times ...
1
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1answer
38 views

Quotient by a discrete subgroup of a Lie group

I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let $H$ be a Lie group and $T$ be a discrete subgroup of its center $Z(H)$. Then there exists ...
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0answers
17 views

Inverse image of a normal subgroup by a group homomorphism is a normal subgroup [duplicate]

Let $f:G\to H$ be a group homomorphism and let $K$ be a subgroup of $H$. If $K$ is a normal subgroup of $H$, show that $f^{-1}(K)$ is a normal subgroup of $G$. Thanks in advance..
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0answers
21 views

Groups with a dense chain of (isomorphic) subgroups

Are there any interesting non-Abelian examples of groups which are equal to the union of a dense chain of normal subgroups, with each subgroup isomorphic to the original group. That is, a group $G$ ...
0
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1answer
30 views

Why is H normal in G equivalent to the conjugate of H by g being a subset of H?

Let $H\le G$ Why are the following equivalent? i) $H \unlhd G $ ii) $ gHg^{-1} \subseteq H$ for all $g \in G$ In other words, why do we not need to show that $H \subseteq gHg^{-1}$ for all $g \in ...
2
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1answer
46 views

Why does $aN=bN \Rightarrow a^{-1}b \in N$?

If $N \triangleleft G$ and $a,b \in G$ then why does $aN=bN$ imply $a^{-1}b \in N$? I don't find it too obvious, sadly, since $N=a^{-1}bN$ by rearrangment, and this tells me that $a^{-1}b$ is the ...
3
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2answers
56 views

A general idea of $G/H$

I am looking t see whether my understanding is correct. SO for a group $G$ and its subgroup $H$, $G/H$ is another group called the quotient group. Now, there are a few things that revolve around this ...
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3answers
41 views

Group theory exercise

Consider the following Group theoretical question: Let $G$ be a group and $H \subset G$ a subgroup. Let $\rho : G \rightarrow H$ be a homomorphism such that the restriction of $\rho$ to $H$, ...
0
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1answer
22 views

proving the identity for subgroups.

What is the best way to prove that if a group is a subgroup of some other group? Or more precisely how to prove that they have common identity element?
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0answers
6 views

Normal subgroups that have only the identity in common commute with each other. [duplicate]

The following is taken from M.A. Armstrong's Groups & Symmetry, #15.6 If $H,J$ are normal subgroups of a group $G$, and if they only have the identity element in common, show that $xy=yx$ for all ...
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1answer
31 views

Coset representatives of principal congruence subgroups $\Gamma_l$ of $SL(n,\mathbb{Z})$

Consider the level $l$ principal congruence subgroup $\Gamma_l$ of the special linear group $SL(n,\mathbb{Z})$ defined as the kernel of the natural map $\phi : SL(n,\mathbb{Z}) \rightarrow ...
3
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1answer
51 views

Subgroups of special linear group SL$(n, \mathbb{Z})$

Are there characterizations of subgroups of a special linear group SL$(n, \mathbb{Z})$? Since SL$(n, \mathbb{Z})$ has infinite order, it would be enough if I know how to generate subgroups of ...
2
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0answers
38 views

Proof that every group of order $15,400$ has a normal subgroup of order $275$

I constructed the groups of order $15,400=2^3\cdot 5^2\cdot 7\cdot 11$ with GAP and noticed that every such group has a normal subgroup of order $275=5^2\cdot 11$. Can this be proven by hand ? ...
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0answers
36 views

if HK is a subgroup for all K, does it imply that H is normal?

we know that in a group G, if H,K be subgroups such that H is normal, then the product HK is also a subgroup. does the converse hold? i.e. if H is a subgroup of a group G such that for any subgroup K ...
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1answer
41 views

Is a union of cosets of a normal subgroup a subgroup itself?

We have a group $G$ and $H$ is its normal subgroup. Say $K = \{C_1,...,C_k\}$ is a subgroup of $G/H$. Is the union C$_1 ∪···∪ C_k$ a subgroup of $G$? I can see that $C_i$’s are elements of the factor ...
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2answers
49 views

Biggest noncommutative group $N$ where a group $G$ is normal.

Given a finite group $G$, it is possible to obtain the biggest non commutative group $N$ such that $G\lhd N$ with $N\neq G$, $\vert N\vert < \infty$ and $N$ not a direct or semidirect product?
2
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1answer
109 views

$G$ contains a normal $p$-subgroup

Let $G$ be a non-abelian finite group with center $Z>1$. I want to show that if $G/Z$ is solvable then $G$ contains a normal $p$-subgroup for some prime $p$ with $p\mid |G:Z|$. $$$$ Since ...
0
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0answers
50 views

Show that $G$ is nilpotent when the quotient group is nilpotent [duplicate]

I want to show that if $H\subseteq Z(G)$ and $G/H$ is nilpotent then $G$ is also nilpotent. $$$$ I have done the following: Since $G/H$ is nilpotent there is a series of normal subgroups $$1\leq ...
0
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3answers
65 views

Isomorphism of Quotient Subgroups

I have a problem on the assignment and I've got stuck at one point. Let $\Bbb{R}$ be the group of real numbers under addition, and let $x\Bbb{Z} ⊂ \Bbb{R}$ be the subgroup which consists of all ...
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2answers
22 views

How to show that there are just $n$ cosets for the $n\mathbb{Z}$ relation?

Consider $n\in \mathbb{Z}$ with $n > 0$, we define $n\mathbb{Z}=\{kn \in \mathbb{Z} : k\in \mathbb{Z}\}$. It is easy to see that $n\mathbb{Z}$ is a normal subgroup of the additive group ...
0
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2answers
57 views

Each minimal normal subgroup of is contained in the center

Let $G$ be finite nilpotent. I want to show that each minimal normal subgroup of $G$ is contained in $Z(G)$ and has order $p$. $$$$ We have that $Z(G)=\{g\in G\mid ga=ag , \ \forall a\in G\}$. ...
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0answers
23 views

Find the normal and the composition series

Could you give me some hints how we could find a normal series and all the composition series of $D_4$ ? $$$$ A normal series of $G$ is $$G\geq G\geq G^{(1)} \geq G^{(2)} \geq G^{(3)} \geq \dots ...
0
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1answer
37 views

Definition normal subgroups

I am slightly confused about the definition of normal subgroups. The book gives the definition that a subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a$ in $G$. Then it explains that one can ...
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1answer
48 views

what does it mean that a subgroup self-normalizes?

I am having difficulty understanding what it means for a subgroup to self-normalize. That is, given $G$ a group, and $H$ a subgroup, $N_G(H)=H$. I've always taken $N_G(H)$ to be a subgroup of $G$ ...
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0answers
19 views

Generating elements of a subgroup

Say, I'm given a group of quanterions $Q_{10}$ under multiplication whose set of elements $\left \{ i,j,k,x,y \right \}$ and I'm asked to generate the elements of the subgroup of $Q_{10}$, say, for ...
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1answer
31 views

Subgroup of a group of bijections

Let $G$ be the group of bijections of the set $X$, which is assumed to have more than two elements. Let $x_0$ ∈ $X$ and $H$ = { $f$ ∈ $G$: $f$( $x_0$)= $x_0$}. Is $H$ normal or not?
0
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1answer
50 views

The group is not simple

I want to show that if $|G|=pqr$ where $p,g,r$ are primes, then $G$ is not simple. We have that a group is simple if it doesn't have any non-trivial normal subgroups, right? $$$$ I have done the ...
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1answer
43 views

Does a group action always induce a quotient?

Let $G$ be a group, let $X$ be a set on which $G$ acts (possibly non-faithfully). I would be tempted to say the following: There exists a normal subgroup $K$ of $G$, such that: $G/K$ acts ...
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2answers
65 views

Can this subgroup be normal with these conditions?

Suppose that $G$ is a group and $H$ is its subgroup such that for every subgroup $K$ of $G$, $HK$ is a subgroup. Can we deduce that $H$ is normal? (I know that the normality of either $H$ or $K$ is ...
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1answer
115 views

A group is isomorphic to the direct product of two subgroups. Conditions for the subgroups to be normal.

Let $(G, \star, \varepsilon)$ be a group and $H$ and $K$ two of its subgroups. Let $$\begin{align*} \diamond\, \colon (H \times K)^2 &\to H \times K \\ \big((h_1,k_1) , (h_2,k_2)\big) & ...
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2answers
32 views

IF a group G of order 8. Then, it is impossible that |Z(G)| = 4 [closed]

I have no idea what equation or proposition I can use to prove that. I know Z(G)≤G.Is that help?
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1answer
18 views

elements in the product of subgroups in $S_4$

Let $N:=\{e,(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3)\}$ be the normal subgroup of $S_4$ and $H:=\langle(1\,2\,3\,4)\rangle$ be the cyclic subgroup of $S_4$ generated by $(1\,2\,3\,4)$. Using the Second ...
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1answer
61 views

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$.

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$. I think this is associated with the action of the left coset of ...
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2answers
47 views

Question about normal subgroups.

If $N$ is a normal subgroup of the finite group $G$ such that the index of $N$ in $G$ and the order of $N$ are relatively prime, show that any element $x \in G$ that satisfies $x^{|N|}= e$ must be in ...
2
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1answer
36 views

Homomorphism and irreducibility

Let $Y: G/N→ GL_d(\mathbb{C})$ be a representation defined by $Y(gN) = X(g)$. Where $N = {\{g\in G: X(g) = I}\}$ is the kernel. Prove that $Y$ is irreducible if and only if $X$ is. attempt: A ...
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1answer
35 views

Intersection of is the kernel of a homomorphism/representation.

Let $\phi: G → GL_d(\mathbb{C})$ be a homomorphism. And let the kernel be $N = {\{g\in G: \phi(g) = I}\}$, and $N $ is a normal subgroup of $G$. Show that for the coset representation , $N = ...
2
votes
1answer
31 views

$ G=H_1\times … \times H_k $ and $ K \unlhd G.$ Then If $K$ is not contained in $Z(G)$, there exists $i$ such that $K\cap H_i \neq \{e\}$

Let a group $G$ be a product of groups, i.e $ G=H_1\times ... \times H_k $ and $ K \unlhd G.$ Then If $K$ is not contained in $Z(G)$, there exists $i$ such that $K\cap H_i \neq \{e\}$. It seems that ...
1
vote
1answer
31 views

Two Subnormal subgroups with index of one and order of other relatively prime

Let $H,K$ two subgroups of a finite group $G$. Suppose that $\gcd(|G:H|,|K|)=1$ Prove that if $K\triangleleft\triangleleft\; G$ then $K\subseteq H$. My idea: Consider before the case $K\triangleleft ...
1
vote
1answer
36 views

Let a finite group $G$ have $n(>0)$ elements of order $p$(a prime) . If the Sylow p-subgroup of $G$ is normal, then does $p$ divide $n+1$?

Suppose $G$ is a finite group and $p$ is a prime that divides $|G|$. Let $n$ denote the number of elements of $G$ that have order $p$ . If the Sylow p-subgroup of $G$ is normal, then is it true that ...
1
vote
1answer
42 views

Give an example for if … [duplicate]

Prove that if $H$ is a normal subgroup of a group $G$ and has index $n$, then for any $g\in G$ we have $g^n\in H$. Give an example for if $H$ is not normal, the mentioned statement is not correct. ...
3
votes
1answer
42 views

Show the kernel of a representation is normal subgroup

Let $X$ be a matrix representation. And let the kernel of $X$ be defined as $N = {\{g \in G: X(g) = I}\}$. A representation is faithful if it's one to one. Show that $N$ is a normal subgroup of $G$ ...
2
votes
2answers
51 views

How can we find all the subgroups?

I want to find all the normal subgroups of $D_n$. We have that $K$ is a normal subgroup of $D_n$ iff $$gkg^{-1}=k\in K, \forall g\in D_n \text{ and } \forall k\in K$$ right? Could you give me some ...