Questions concerning normal-subgroups of groups.

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-4
votes
2answers
45 views

How can I find all the numbers of order 10 in $Z_{60}$? [on hold]

How can I find all the numbers of order $10$ in $Z_{60}$? In fact, these are all the numbers with $(x,60)=6$. How can I find all these numbers?
-3
votes
0answers
24 views

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? [on hold]

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? If yes, so hwo can i prove it? [without Sylow Theorems] p,q > 1
0
votes
1answer
25 views

Normal subgroup. [duplicate]

Let $N$ be subgroup of a group $G$. Suppose that, for each $a\in G$, there exists $b\in G$ such that $Na=bN$. Prove that $N$ is a normal subgroup. Please guide me with a proof. Thank you for your ...
2
votes
0answers
39 views

To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$ \bigcap_{N \in \mathcal{N}} N \ = \ \{ e\} $$ I know how free ...
0
votes
1answer
29 views

Find the Quotient group of normal subgroup of order 4 of quaternion group.

$H=(\pm 1 , \pm i , \pm j, \pm k)$. I know $$c(i)=( \pm 1, \pm i) \quad c(j)=( \pm 1, \pm j)\quad c(k)=( \pm 1, \pm k) $$ and thus the class equation of $H$ is $8= 2+2+2+2$ because $\lvert H\rvert =...
0
votes
0answers
20 views

finding total no. of subgroups of a group

How do we find the total no. Of subgroups of a group, in Zn,Sn,An,Dn? And there's this theorem that if 'a' belongs to Group G,then is a subgroup of G. Does that mean that we already hve available ...
2
votes
3answers
91 views

If $G/N\cong H$ then $G=NH$?

I am curious if $G/N\cong H$ then $G=NH$? ($N$ is a normal subgroup of $G$, and $H$ is a subgroup of $G$.) With this setup, we get that $NH$ is a subgroup of $G$ so $NH\subset G$. I am not sure ...
1
vote
2answers
38 views

$\mathbb{Z}$ has no composition series. Need an assistance in some questions.

Please, read the whole post before trying to answer. Remark: here $\subset$ means "strict inclusion". I need to prove that the group $\mathbb{Z}$ has no composition series. That is, no normal series ...
0
votes
2answers
33 views

$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?

Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ ...
1
vote
3answers
33 views

Finding all normal subgroups of $A_4$

I was reading up on this: Find the number of normal subgroups of $A_4$. If $H$ has a $3$-cycle, say $(123)$, then $H$ has its inverse $(132)$ thefore it also has $(124) = (324)(132)(324)^{-1}$, ...
1
vote
1answer
32 views

If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$

Let $\varphi: G \longrightarrow G'$ be a homomorphism group. If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$. Of course the Ker $\varphi$ and Im $\...
2
votes
3answers
82 views

Index of intersection of two normal subgroups

Question: Let $G$ be a finite group and $H$ and $K$ are normal subgroup of $G$. If $[G:H]=2$ and $[G:K]=3$, determine $[G:H∩K]$. My attempt: I think $[G:H∩K] = 6$. Since $G$, $H$, $K$ are ...
2
votes
1answer
46 views

If a subgroup has smallest prime index, then it is normal

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...
0
votes
2answers
35 views

Finding groups $H$ for which there exists surjective homomorphisms $f:D_4 \rightarrow H$?

How can I find out for which groups $H$ there exists surjective homomorphisms $f: D_4 \rightarrow H$? $D_4$ is the dihedral group of the square. I have a theorem that says that there exists such ...
2
votes
2answers
46 views

Group of order $54$ has normal sugroup of order $27.$

Let $G$ be a group of order $54$. Prove that there exists a normal subgroup of order $27.$ Is this normal subgroup unique? Thoughts. Since $27$ divides $54$, by Lagrange's theorem we can not exclude ...
0
votes
3answers
61 views

If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$

I wish to prove whether this is true or false. If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$. I'm not even sure if $N$ being normal ...
3
votes
2answers
57 views

We Quotient an algebraic structure to generate equivalence classes?

Till now I visualized Quotient groups as a technique to generate equivalence classes whenever needed, as in $\mathbb{Z}/n\mathbb{Z}$. But now I have a feeling of doubt since I haven't seen a book that ...
2
votes
3answers
34 views

Let $G=\langle \mathbb{Z},+\rangle $ and $H=\{6n|n \in \mathbb{Z}\}$. Find all the distinct left and right cosets of $H$ in $G$.

I have an exercise where I am supposed to find the left and right cosets. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set $H$ and ...
0
votes
1answer
41 views

$H\cap K$ is normal maximal subgroup of $H$.

if $G$ is a group and $H$ and $K$ are different normal maximal subgroup of $G$ .Prove that: $H\cap K$ is normal maximal subgroup of $H$. If we suppose that $L$ is subgroup of $G$ which $H\cap K \...
1
vote
0answers
10 views

Normalizer of some subgroup of a group is necessarily normal subgroup of the group. [duplicate]

Let $G$ be a group and $H$ be subgroup of $G$. I already know that normalizer of H in G $N_G(H)$ is the largest subgroup of $G$ having H as a Normal subgroup. So I showed that $N_G(N_G(H))=N_G(H)$. ...
2
votes
2answers
16 views

Normal subgroup of a finite group of order 21 [duplicate]

How many normal subgroups does a non abelian group $G$ of order 21 have other than the identity subgroup $\{e\}$ and $G$? I think we can solve the problem using syllow's $p$-subgroups
0
votes
0answers
12 views

Normal subgroup related

let $G$ be a finite group and $N$ be the normal subgroup of $G$ of index $p$, a prime number. Then what is number of all subgroups of $G$ containing $N$? I don't have idea
1
vote
2answers
28 views

suppose that $H \leq G$ and $N \unlhd G$ and $(|H|,|G:N|) = 1$ then prove that $H \leq N$

suppose that $H \leq G$ and $N \unlhd G$ and $(|H|,|G:N|) = 1$ then prove that $H \leq N$ I don't know how to start ! I know that I should prove that $H \subseteq N$!Can N be infinite subgroup?Can ...
1
vote
1answer
26 views

Proving that if a finite group $G$ has a composition series of length 2, then any composition series has length 2

I'm having trouble proving this. If $G$ has a composition series of length 2, then that means there is a maximal normal subgroup $H$ of $G$ such that $H$ is simple, i.e. we would have a composition ...
1
vote
1answer
35 views

Minimal field extension of $\mathbb{Q}(\sqrt[6]{2})$ over $\mathbb{Q}$ and Galois group

Find a minimal field extension $L$ of $\mathbb{Q}(\sqrt[6]{2})$ such that $L$ is normal over $\mathbb{Q}$ $L$ is normal over $\mathbb{Q}$ which means that it is the splitting field of polynomials in ...
0
votes
1answer
25 views

Given homomorphism $\rho: G \times H \to G$, trouble verifying Homomorphism Theorem

I've just run into a little confusion when doing questions on the Homomorphism Theorem. Say we have a homomorphism $\rho: G \times H \to G$ such that $\rho(g,h) = g,$ where $G \times H$ represents ...
1
vote
2answers
59 views

Normal subgroups of matrices

Let $G=\begin{bmatrix}1&a\\0&b\end{bmatrix}$ so that $a,b\in\mathbb C$ and $b\ne0$. I need to prove that $G$ has infinitely many normal subgroups. I attempt to do this by constructing some ...
0
votes
1answer
19 views

Showing that a subnormal series for a finite group $G$ can be made into a composition series for $G$

Suppose that $\{e\} < G_1 < G_2 < \cdots < G_n = G$ is a subnormal series, thus for all $i$, we have $G_i$ is normal in $G_{i+1}$. How can I show that this can be "refined" to a ...
1
vote
0answers
23 views

A surjective homomorphism and normal subgroups [duplicate]

I'm reviewing group theory for a comprehensive exam and this question came up. Suppose I have two groups $G$ and $K$ and $\varphi$, a surjective homomorphism from $G$ to $K$. How can I prove that ...
1
vote
2answers
116 views

Showing that a quotient group $G/N$ is isomorphic to $\mathbb{Z}_3$

I have permutations $\sigma=(135)(27)$, and $\tau = (27)(468)$. $G =\langle \sigma,\tau \rangle$ and $N$ is the smallest subgroup of $G$ that contains $\tau$, so $N = \langle \tau \rangle$. $|\sigma| =...
3
votes
1answer
55 views

Determing whether a subgroup is normal

I have been working with normal subgroups and feel like I am doing something wrong. I understand there are many ways to demonstrate if a subgroup is normal, but the methods seem to take longer than I ...
0
votes
2answers
47 views

$S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. $...
0
votes
1answer
33 views

If the correspodence $aHbH=abH$ defines a group operation on the set of left cosets of $H$ in $G$, then show that $H$ is normal in $G$.

If the correspodence $aHbH=abH$ defines a group operation on the set of left cosets of $H$ in $G$, then show that $H$ is normal in $G$. My attempt: Let $x\in G$. Then we know that $xHx^{-1}H=H$. ...
1
vote
2answers
49 views

A group of order pq with a single subgroup of order p [closed]

Given a group $G$ of order $pq$ (such that $p,q$ are primes and $p < q$) that have a single subgroup of order p (named $H$) prove that $\forall h \in H , g\in G : ghg^{-1} = h$
0
votes
1answer
28 views

Show that if $K$ is a normal subgroup of $H$ such that $H/K$ is abelian, then K contains all of the 3-cycles.

Claim: For $n≥5$, if $H$ is a subgroup of $S_n$ which contains all of the 3-cycles, and $K$ is a normal subgroup of $H$ such that $H/K$ is abelian, then K also contains all of the 3-cycles. Attempt: ...
2
votes
1answer
33 views

How to find normal subgroups from a character table?

I know that normal subgroups are the union of some conjugacy classes Conjugacy classes are represented by the the columns in a matrix How could we use character values in the table to determine ...
0
votes
0answers
24 views

If $H$ and $K$ are nilpotent normal subgroups then $C(HK)$ is non trivial

I know that this follows from the fact that $HK$ is nilpotent but maybe there is an easier way to proof this? I wanted to show that there is an Element in $HK$ that commutes with $H$ and $K$. I ...
2
votes
2answers
51 views

Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
1
vote
1answer
38 views

Groups of order 12 with a normal 3-subgroup contain an element of order 6

Let $G$ be a group of order $12$ with a normal $3$-subgroup (which is unique by Sylow's theorems). Does it contain an element of order $6$? I just need a hint to prove it without classifying all the ...
-1
votes
1answer
24 views

If the intersection of two normal subgroups is trivial, then their elements commute [closed]

How to show that if $N \ \& \ M$ are 2 normal subgroups of group $G$ and $N\cap M=\{e\}$ (identity element), then for any $n\in N \ \&\ m\in M $, $nm=mn$?
2
votes
2answers
76 views

All subgroups normal $\implies$ abelian group

This is , I think an easy problem just that I am not getting the catch of it. How to show whether or not the statement is true? All subgroups of a group are normal$\implies$ the group is an abelian ...
0
votes
5answers
37 views

Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic

I need some help on this one. $G$ is a group. If $N$ is a subgroup of $G$ contained in the centre of $G$ and $G/N$ is cyclic, show that $G$ is Abelian. My attempt is only half way and stuck at ...
0
votes
0answers
46 views

A Group G is abelian $\Leftrightarrow$ $ Inn(G)$ is a normal subgroup of Sym(G)

First of all I don´t think that this question is answered here If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$ because in my opinion ...
0
votes
0answers
24 views

Criterion for a group $G$ to be abelian using $Inn(G)$ [duplicate]

Let $(G, *)$ be a group. Let $\mathcal S(G)=\{f:G\to G\space|f\space is\space bijective\}$ be the symmetric group and $Inn(G)=\{\kappa_{a}\space|\space a\in G\}$ the inner automorphisms. Now I want to ...
1
vote
2answers
28 views

Let $N_i \subseteq G_i$ for $i = 1, …, n$ be Normal subgroups. Show that $\prod_{i=1}^{n}N_i \subseteq \prod_{i=1}^{n}G_i$ is a Normal subgroup

a) Let $N_i \subseteq G_i$ for $i = 1, ..., n$ be Normal subgroups. Show that $\prod_{i=1}^{n}N_i \subseteq \prod_{i=1}^{n}G_i$ is a Normal subgroup b) Find an isomorphism $\prod_{i=1}^{n}G_i / \...
2
votes
2answers
57 views

Normal subgroups of $A_5$ must contain a 3-cycle.

I am trying to prove the simplicity of $A_5$ by showing that every non-trivial normal subgroup $H$ contains a 3-cycle, and therefore is all of $A_5$ since the 3-cycles all belong to one conjugacy ...
1
vote
3answers
43 views

Two normal subgroups and isomorphism theorem

Question Let $N_1$ and $N_2$ be normal subgroups of $G$. Prove that $N_1N_2/(N_1\cap N_2) \cong (N_1N_2/N_1)\oplus (N_1N_2/N_2)$. I think the homomorphism must be $\phi : N_1N_2 \to (N_1N_2/...
0
votes
3answers
63 views

Are all normal subgroups Abelian?

If $H \subset G$ is a normal subgroup of G, => $xHx^{-1} = H$ or $xH = Hx$ for all $x \epsilon G$ => $xH = Hx$ for all $x \epsilon H$ Hence, all normal subgroups of a group are themselves Abelian? ...
0
votes
1answer
26 views

Order $4$ subgroup of alternating group $A_4$

I ran into the following problem: Let $H$ be the subgroup $H = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ in $G = A_4 = H \cup \{(1\, 2\, 3), (1\, 3\, 2), (1\, 2\, 4), (1\,4\,2), (1\,3\,4),(...
0
votes
1answer
25 views

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

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