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Questions tagged [normal-subgroups]

For questions concerning normal subgroups of groups.

3
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2answers
60 views

How do i prove that any subgroup of $A_5$ has order at most 12? [duplicate]

I know this question has been answered Other proofs that subgroups of $A_5$ have order at most 12 But i have difficulty in understanding that proof.The book says we can assume that $A_5$ has no ...
0
votes
0answers
37 views

Intersection of normal subgroups is normal [duplicate]

Can someone tell me if this proof is sufficient given the assumption. Let H,K be normal subgroups of G. Assumption: g(H∩K)= (gH)∩(gK) Proof: g(H∩K) =(gH)∩(gK) =(Hg)∩(Kg) =(H∩K)g ...
8
votes
0answers
66 views

Is there a categorification of “(virtually) solvable”?

If this question doesn't make sense or is otherwise poor quality, then I'm sorry. Motivation: As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable ...
-4
votes
0answers
37 views

Prove that $A_5$ has no normal subgroups $N$ $≠$ $e$ , $A_5$? [duplicate]

$A_n$ is subset of $S_n$ consisting of all even permutations. I have read the permutation groups theory but I don't know how to solve this problem (I mean how to proceed ). I want little simple ...
0
votes
0answers
42 views

Normal subgroups are not transitive, but $H \subseteq N \lhd G, H \lhd G \implies H \lhd N$?

Let $H,N,G$ be groups where $H \subseteq N \subseteq G$. For the 3 statements below: $H$ is a normal subgroup of $G$: $ghg^{-1} \in H \ \forall h \in H, \forall g \in G$ $N$ is a normal subgroup of $...
2
votes
1answer
29 views

Showing that a conjugacy class is completely contained in a normal subgroup

I am working on a homework problem, and although I do not want a solution, I want to ask what is wrong with a certain approach I have. If $H\subseteq G$ is a normal subgroup in $G$, and $[g]_G$ is ...
3
votes
1answer
54 views

relation of certain definitions for groups

I have the following definitions which I want to compare: Definition 1: Let $G$ be a group and $H$ be a subgroup. $H$ is called normalish if for any finite sequence $g_1,\dots,g_n\in G$ the ...
0
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0answers
28 views

Using the first isomorphism theorem to prove subgroups

How can I prove that if $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, then $HN$ is a subgroup of $G$ using the first isomorphism theorem? I know how to prove this in general, but my ...
1
vote
1answer
16 views

Normal subgroup generated by $D_8$

Let $D_8 = <a,b : a^4 = b^2 = 1, bab^{-1} = a^{-1}>$ be the dihedral group. I'm trying to show that the subgroup generated by $a^2$ is normal. But, isn't $<a^2> ={\{1, a^2}\}$? So the ...
2
votes
1answer
45 views

What's an example of a finite, non-abelian, non-simple group that is *not* semidirectly reducible? [duplicate]

Say I want to classify all groups of a given order. The abelian case is completely understood by the structure theorem for finitely generated abelian groups. Assume our group is non-abelian, and we ...
1
vote
2answers
31 views

Existence of normal subgroup of group with given property

Let G be a finite group in which $(ab)^p=a^pb^p$ for every a,b in G where p is prime dividing order of G Then if P is p- Sylow subgroup of G then there exists a normal subgroup N of G with $P\cap N=...
0
votes
1answer
11 views

Find a nontrivial, proper, normal subgroup of $S_5⊕D_4$ [closed]

I can't find a normal proper nontrivial subgroup for this. I only need it to start a proof.
1
vote
1answer
29 views

If $H$ and $K$ are normal subgroups of $G_1$ and $G_2$, Prove that $H\oplus K$ is a normal subgroup of $G_1 \oplus G_2$

Let $G_1, G_2$ be groups and $H\leq G_1$ and $K\leq G_2$. This is what I have so far: Let $x \in H \oplus K$ then $x=(h,k)$ for some $h \in H$ and some $k \in K$. $H\leq G_1$ therefore, $h\leq G_1$. $...
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0answers
9 views

left translation

how can i prove that if $N$ is a closed subgroup of $G$ with a right translation , then is a left translation invariant $W*$- subalgebra of $L$ infinity $G$? https://projecteuclid.org/download/...
5
votes
1answer
34 views

Equivalent to subgroup being normal

From Elements of Abstract Algebra question 46$\beta$: Show that a subgroup H of a group G is normal if and only if ab $\in$ H implies $a^{-1}b^{-1}\in$ H for any elements a,b $\in$ G Showing left ...
2
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0answers
37 views

Are two lifts of a generator of a cyclic quotient conjugated?

Let $G$ be a group and $I$ be a normal subgroup of $G$ such that $G/I$ is cyclic, and let $\bar{g}$ be a generator. Let $g, g' \in G$ be two lifts of $\bar{g}$, i.e. $g + I = g' + I = \bar{g}$. ...
0
votes
1answer
43 views

Proof that $\langle y\rangle$ is a normal subgroup of $G$ with order $q$ and $|G|=pq$ with $p$,$q$ prime

Let $p, q$ prime, with $p<q$ and $G$ a group of order $pq$. Then by Cauchy's theorem $G$ contains elements $x$ and $y$ of order $p$ and $q$ respectively. I have already proven that $\langle x, y\...
0
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0answers
32 views

Show that a non-abelian group of order 8 with a single element of order 2 is isomorphic to Q group

where Q is the quaternion group. Thanks ! Here are my thoughts : With the Lagrange's theorem, we have that the order of the subgroup divides the order of the group. But there is a single element ...
0
votes
3answers
48 views

Find normal subgroup of the following group

Is the group $G$ given by $$\left\{\begin{bmatrix} 1 & \alpha &\beta \\0& 1 &\gamma\\0 &0 &1\end{bmatrix}:\alpha,\beta,\gamma \in \Bbb R\right\}$$ simple? My try: Obviously ...
0
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0answers
11 views

Intuitive explanation for: let $I_G$ be the group of > all inner automorphisms of $G$. Then $I_g$ is isomorphic to $G/C_G$

In the book of Fundamental Concepts of Abstract Algebra by G. Ehrlich, at page 106, it is given that Let $G$ be a group with centre $C_G$, and let $I_G$ be the group of all inner automorphisms of ...
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0answers
26 views

Let $K \subseteq G$. Nec. and suff. cond. for that $\exists$ a normal subgr $H$ of $G$ s.t $exists$ transversal $I$ s.t $I \subseteq K$.

I am trying to solve/understand the following question that I have come up with; Let $G$ be a group, and $K \subseteq G$ be given. What are the necessary and sufficient condition for that there ...
4
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1answer
61 views

If $H$ and $K$ are normal subgroups of $G$ and $H\bigcap K = \{e\}$, prove that $G$ is isomorphic to a subgroup of $G/H \times G/K$

I tried proving this in the following manner, but I am not confident with these types of problems so any verification would be appreciated. Thank you. Let $A = \{(gH, gK): g \in G\}$ Define $\phi$ : ...
1
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1answer
33 views

Multiplication of Subgroups and Internal Direct Products

Challenge problem on my Abstract Algebra Homework that I was unable to make any significant progress on. Any hints for parts (a) and (b) would be much appreciated. It seems that one direction for part ...
0
votes
2answers
34 views

Why require a normal subgroup to be also a subgroup, in the definition?

Let $G$ be a group, $H\subset G$ (but not necessarily $H \leq G$!) and, for all $g\in G$, $gHg^{-1}\subset H$. Is it true that $H$ is a subgroup of $G$? (And if it happens to be true, why to require a ...
2
votes
0answers
22 views

Question about Quotient of Product of Subgroups

Let $G$ be a group, $H, K, N$ be subgroups such that $N$ is normal in $G$ and $K$ is normal in $H$. I would like to know whether there exists an injective homomorphism from $\frac{NH}{NK}$ to $\frac ...
0
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1answer
22 views

Proving the bound $[G:Core(H)]\leq [G:H]!$

I was having trouble figuring out how exactly to prove $[G:Core(H)]\leq [G:H]!$ where G is a group and H is a subgroup of G. I know Core is the kernel of the homomorphism from G to $S_{G/H}$ induced ...
0
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2answers
42 views

Prove that $\varphi\big(C_G(x)\big) =C_H\big(\varphi(x)\big)$, where $\varphi:G\to H$ is a group homomorphism with certain properties.

Let $N$ be normal in $G$ and suppose that $\varphi :G \to H$ is surjective group homomorphism such that $N \cap \ker(\varphi) =1$. Show that $\varphi\big(C_G(x)\big) =C_H\big(\varphi(x)\big)$. I am ...
0
votes
1answer
22 views

Let $P$ be a partition of a group with $AB \subseteq C$. Why is $1 \in P_n$? $P_n$ is the equivalence class of $n \in N$ and $1 \in N=P_1$.

Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be ...
0
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1answer
48 views

Partition of a conjugacy class to conjugacy classes of a normal subgroup

Let $G$ be a group, $H$ be a normal subgroup of $G$, and $O$ a conjugacy class of $G$ contained in $H$. Consider $O = \cup_{i = 1}^{n}O_i$ the partition of O into conjugacy classes of $H$. Show that ...
0
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1answer
26 views

In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?

There is an answer here, but it is a "roadmap". group containing normal subgroups of orders $3$ and $5$ contains element of order $15$ There are answers here, but they are "roadmaps" too. If $G$ ...
1
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1answer
35 views

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

In this question Prove that the product $NK$ of two normal subgroups $N$ and $K$ of a group $G$ is a normal subgroup of $G$, and $NK=KN$., it is proved that $NK=KN$ if both $N$ and $K$ are normal ...
6
votes
1answer
50 views

Validation for a conjecture about Chinese Remainder Theorem for groups

I was wondering if the following statement is true: Let $G$ be a group with normal subgroups $H_1,H_2,...H_n$. Suppose $H_iH_j=G$ for all $i\neq j$. Then $G/H_1\cap H_2...\cap H_n\cong G/H_1 \times......
0
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0answers
20 views

Discrete normal subgroup of a connected linear lie group

It is known that a discrete normal subgroup N of the group G is contained in the center of G. But we also know that if a group is discrete then its lie algebra is {0}, and we know that if something is ...
1
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1answer
46 views

Prove the number of conjugates of K in L is $[Gal(L|F) : N]$ where $N$ is the normalizer subgroup

Let $F ⊂ L$ be Galois and $F ⊂ K ⊂ L$ be an intermediate field. Given $σ_1K, · · · , σ_rK$ are the distinct conjugates of $K$ where $σ_1 = e$ and $σ_j ∈ Gal(L|F), 1 ≤ j ≤ r$, I wish to prove $r = [Gal(...
0
votes
1answer
57 views

>$f : G → G′$ is a group homomorphism. (i). $H′$ is a normal subgroup of $G′$. Prove $f^-1(H′) = [ h ∈ G|f(h) ∈ H′] $ is a normal subgroup of $G$.

$f : G → G′$ is a group homomorphism. (i). $H′$ is a normal subgroup of $G′$. Prove $f^{-1}(H′) = [ h ∈ G|f(h) ∈ H′] $ is a normal subgroup of $G$. I don't know how to approach this question. ...
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2answers
41 views

Prove that if a subgroup $H$ of $G$ contains all of its conjugates, it is a normal subgroup.

It is known the conjugate closure of a subgroup $H$, $H^G$, contains $H$. But the conjugate closure can be interpreted as the union of all the conjugates of $H$. The question is If it is given for ...
0
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0answers
36 views

Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$ then $G/H \cong K?$

Determine whether the following is true or false. Note that $\cong$ means group isomorphic. Question: Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$...
0
votes
0answers
39 views

Show $\phi:H\to HN/N$ is one-to-one.

Can anyone help me with the question? I am stuck on showing this is one-to-one. For subgroup $H$ of $G$ and normal subgroup $N$ in $G$, show the function $\phi:H\to HN/N$ defined by $\phi(h)=hN$ is ...
0
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0answers
16 views

Defining a normal subgroup to be maximal.

I just want to clear up something from Question 2. from the following my professor made: For Question 2., if $|G|=p$ and $N$ is maximal, does that mean $|N|\neq p$ since there is no subgroup $H$ of $...
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1answer
42 views

Show that $\phi$ induces a canonical homomorphism.

Can someone give me a hint on how to solve the following question from the textbook Abstract Algebra: Theory and Applications by Thomas W. Judson? Do I just need to show that $\bar{\phi}$ is a ...
0
votes
2answers
23 views

Even and odd elements in more general additive groups

My question is motivated by this paper in whose second page the author describes an analog of even and odd numbers for any additive group $G$ with a subgroup of index $2$, denoted $G^+$: the elements ...
0
votes
1answer
47 views

Does $P^g=\{gpg^{-1}|g \in G\}$ have the same order as P, where |P|=7

Say we have a group G, s.t P is a subgroup of G and |G|=28,|P|=7. then consider $P^g=\{gpg^{-1}|g \in G\}$ what is the order of this group ? I think that it is seven as well because if we consider the ...
1
vote
1answer
28 views

Normal closure of a subgroup that is identical to its commutator subgroup inherits this property

Problem statement Let $A$ be a subgroup of $G$. It is known that $A = A^{(1)}$, where $A^{(1)}$ is $A$'s commutator subgroup. Normal closure of $A$ is defined as $N(A) := \left\langle g A g^{-1} \ | ...
0
votes
1answer
16 views

What does normality is image closed mean here?

So I $K,H$ are normal subgroups of G and K is a subgroup of H. Then H/K is normal in G/K. The argument given was "because normality is image closed under quotient map by K normal subgroup of H and G ...
0
votes
1answer
30 views

Every finite group has a a sequence of nested subgroups where adjacent elements are normal [duplicate]

A group $Q$ is called simple if $|Q|>1$ and the only normal subgroups of $Q$ are the trivial subgroups $\{e\}$ and $Q$. Prove that for any finite group $G$ there exists a sequence of nested ...
0
votes
0answers
48 views

Normal Fuchsian subgroup of $PSL(2,\mathbb{R})$

I've been working with Fuchsian groups and from geometrical motivations finding a cocompact normal Fuchsian subgroups of $PSL(2,\mathbb{R})$ would have intresting properties for my research. It is ...
2
votes
1answer
39 views

Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective.

I am trying to show the following: Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective. Note that $H\leq G$ and $N$ is normal in $G$. My attempt so far: Let $\phi(h_{1})=\phi(h_{2})$...
3
votes
4answers
261 views

Can a group have two different subgroups of index $2$?

I know that subgroup of index $2$ is normal. I am interested in knowing that is that subgroup is unique or there exist example of subgroup which can have two subgroup of index $2$? If there is ...
1
vote
1answer
33 views

The normal subgroup of G generated by the set A

Let $G$ be a set and let $A \subseteq G$. Then the normal subgroup of $G$ generated by the set $A$ is $$\langle A \rangle ^N= \{ g_1a_1^{i_1}g_1^{-1} \dots g_ma_m^{i_m}g_m^{-1} \mid m \geq 0, a_j \in ...
2
votes
0answers
61 views

Ambiguity of the notation $\lhd$

If a group $G$ have two subgroups $A,B$ which are isomorphic. Can I say that $A\lhd G$ if $B\lhd G$? Indeed we often do in this way when we don't considerate about the concrete set of groups.