Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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2
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2answers
72 views
0
votes
0answers
78 views

A regarding of state vector

A state vector X for a four-state Markov chain is such that the system is four times as likely to be in state 3 as in 4, is not in state 2, and is in state 1 with probability 0.2. Find the state ...
2
votes
2answers
150 views

How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
0
votes
1answer
39 views

A question about normal subgroups of nilpotent group

Assume G is a nilpotent group and to any n dividing $|G|$, if there is always a normal subgroup of G with order n?
5
votes
2answers
74 views

Show that $f$ is a homomorphism.

There is a group $G$ of order $p^3$, where $p>2$. Show that $f:G\rightarrow Z(G) $ with $f(x)=x^p$ is a homomorphism. My attempt: Case a): Suppose $|Z(G)|=p^3$. Then $G=Z(G)$, so $G$ is abelian, ...
-2
votes
1answer
77 views

topic between algebra and geometry [closed]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
2
votes
0answers
112 views

Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
1
vote
1answer
35 views

Let $ O_{p^{\prime}}(G/A) = T/A $, Why $ T \leq F $ and $ [A , T]=? $

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
0
votes
1answer
37 views

existence of a subgroup of a solvable group G of order d for any divisor d of |G|, with d<|G|^(1/2)

Let $G$ be a solvable group of order $n$ and $d<\sqrt{n}$ be any divisor of $n$. Is there any subgroup of $G$ of order $d$?
2
votes
0answers
35 views

Let $ F = VN $ that $ V \cap N = 1 $ . Let $ L = N_{G}(V) $. $ (\vert N \vert , \vert F/N \vert) = $?

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
6
votes
2answers
100 views

Cubic Planar Graphs have $2^m-1$ Hamilton Cycles, contradicting Bosak…

I looked at the symmetric difference of hamilton cycle (HC) in cubic planar graphs and found that, together with the empty graph, they build a subgroup of the abelian group $\Omega$ of symmetric ...
0
votes
1answer
36 views

Let N = Fit(G). Why $ N = O_{p}(G) $ and $ A \leq Z(N) $?

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
4
votes
1answer
43 views

Presentations of the unity group

I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for ...
1
vote
4answers
82 views

Prove that the 4-group V is normal subgroup of $S_4$ by using isomorphism theorem

Prove that the 4-group V is normal subgroup of $S_4$ First, by using the multiplication table, I am able to prove that 4-group V is subgroup of $S_4$. But I face problem in proving that $\forall x\...
1
vote
2answers
58 views

Computing Factor Group step-by-step manually

I am reading Fraleigh's Abstract Algebra $\S$15 on factor group, Example #15.11: Compute factor group $(\mathbb Z_4 \times \mathbb Z_6) / \langle (2, 3)\rangle.$ The text gives a solution that ...
2
votes
0answers
23 views

Find the intertwiner of two equivalent group representations

Let $G$ be a finite group. Let $$\{A(g)\mid g \in G\} \text{ and }\ \ \ \{B(g) \mid g \in G\}$$ be two equivalent unitary matrix representations of $G$. How do I find a unitary intertwining matrix $S$...
0
votes
0answers
27 views

$ G $ is supersoluble if and only if every normal subgroup of $ G $ with index no more than 2 satisfies the maximal permutizer condition.

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
0
votes
1answer
37 views

Prove that if $H$ is a unique subgroup of order $|H|$ then $H$ is a characteristic subgroup.

Let $G$ be a group and $H\leq G$. $H$ is a unique subrgroup of order $|H|$. Prove that $H$ is a characteristic subgroup. I tried to show by contradiction that if $\phi (h)\not \in H$ then I can find ...
1
vote
0answers
28 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
1
vote
2answers
38 views

$H\subseteq G$, $N\triangleleft G$, and showing $|[G:N]|$ is a prime number

Let $G$ be a finite group and let $N\triangleleft G$ a normal subgroup. It is given that if $H\subseteq G$ such that $N\subseteq H\subseteq G$, then $H=N$ or $H=G$. Show that $|[G:N]|$ is a prime ...
5
votes
1answer
104 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq s,t\...
0
votes
0answers
42 views

For which integers $r$ is $\sigma ^r$ also a $k$-cycle? [duplicate]

Let $\sigma$ be a $k$-cycle in $S_n$. For which integers $r$, is $\sigma ^r$ also a $k$-cycle? I think I managed to prove that this is true iff $(k,r)=1$, but my proof was too long and not elegant ...
2
votes
1answer
118 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
0
votes
1answer
35 views

Prove that if $a\in G^n$ and $g\in G$ then $g^{-1}ag\in G^n$

Let $G$ be a finite group and $n\in \mathbb{N}$. For all $a,b\in G$ there is: $(ab)^n=a^nb^n$. Define $G^n=\{g^n\ |\ g\in G\}$. Prove that $G^n$ is a subgroup of $G$ and that if $a\in G^n$ and $g\...
2
votes
1answer
36 views

A question on finite abelian groups

Let $m_1,\dots,m_k$ be positive integers. Are there positive integers $d_1,\dots,d_k$ such that $d_i|d_{i+1}$ and $$ \oplus_{i=1}^k \mathbb{Z}/m_i\mathbb{Z}\cong \oplus_{i=1}^k \mathbb{Z}/d_i\mathbb{...
4
votes
4answers
244 views

Find the order of an element of finite group

Let $G$ be a finite group and $g,h\in G-\{1\}$ such that $g^{-1}hg=h^2$. In addition $o(g)=5$ and $o(h)$ is an odd integer. Find $o(h)$. I know from a previous exercise that if there exists a ...
6
votes
2answers
245 views

How to tell if a given equation is not a class equation of a group?

Which of the following cannot be a class equation of a group of order $10$? $1+1+1+2+5=10$ $1+2+3+4 =10$ $1+2+2+5 =10$ $1+1+2+2+2+2=10$ As I can see options 2, 1 and 4 are not ...
1
vote
1answer
68 views

What groups are that? What does : mean?

What are the groups 2^6 : 3 . S_6 or 2^4 : A_8 ? Are they some subgroups of S_6 or A_8? I believe that 2 . A_n is the double cover of A_n, and "multiplying" with a number gives a covering group. But ...
0
votes
1answer
37 views

criteria for a short exact sequence of finite groups to be split

Suppose you have a short exact sequence of finite groups $1\rightarrow N\rightarrow F\rightarrow G\rightarrow 1$ such that $|G|$ and $|N|$ are coprime. Must the sequence be split? (Here I mean the ...
3
votes
5answers
272 views

$H,K$ are normal in $G$, then $HK$ is normal in $G$ (product of normal subgroups is normal)

This is a proof I couldn't find anywhere. Could somebody give me a help? I need this to show that $$\frac{H}{H\cap K}\cong \frac{HK}{K}$$ but to form the quotient group I need first to show that $H\...
1
vote
1answer
47 views

Why does the Principle of Well-Ordering imply a remainder of $0$ for the division algorithm?

I'm currently reading a text (Thomas W. Judson, Abstract Algebra - Theory and Applications) where the author proofs the theorem that every subgroup of a cyclic group is cyclic. The proof goes as ...
0
votes
3answers
122 views

Finding a normal and not normal subgroup of $S_3$

I'm being asked to find 2 subgroups of $S_3$, one of which is normal and one that isn't normal. I guess, to find the non normal subgroup is easier. I would do this by trial and error, but since the ...
2
votes
1answer
115 views

Homomorphism of Free Groups

I am reading the theorem of homomorphism of free group from Fraleigh's text in $\S$36 and could only get a fuzzy idea at best: Let $G$ be generated by $A = \{a_i \mid i \in I \}$ and let $G'$ be ...
0
votes
1answer
74 views

Do cycle graphs determine groups up to isomorphism?

This erroneous version of the wikipedia article for Cycle Graphs stated that the cycle graph of different groups could be the same. The immediate error is that it stated that the cycle graph of $\...
2
votes
1answer
73 views

Confusion with Centers, Conjugacy Classes, and Normal Subgroups

Ok this is a bit of a soft question, but I am in my first semester of algebra and getting continually confused by (what seems to me) some competing sets. Let $G$ be a group The center of $G$ is $...
3
votes
0answers
45 views

Relations between the Latin squares of order n and the groups of order n.

Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. ...
4
votes
1answer
108 views

Is this parabolic induction?

In one of my previous questions, @PL. explained the idea of parabolic induction. In my bachelor thesis I used a technique quite like this, to find all irreducible representations of a certain group. I ...
4
votes
2answers
92 views

Showing Sylow $p$-groups are Abelian if $n_p=2p+1$

An old qual question on Sylow $p$-groups: Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian. ...
0
votes
0answers
25 views

What is the purpose of the almost maximal and $ p $-supersoluble subgroup?

Suppose that $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose that there is an element $ y \in H $ such that $ H = \langle y \rangle L $ for any almost maximal subgroup L of $ H $; then $ G $ ...
5
votes
1answer
62 views

How do we identify $\mathfrak{R}$-automorphisms of a group?

If $G$ is a finite group, a bijection $f\colon G\to G$ is called a (normed) $\boldsymbol{\mathfrak{R}}$-automorphism if $f$ maps subgroups of $G$ to subgroups of $G$, and $f(gH) = f(g) f(H)$ for any ...
1
vote
2answers
143 views

Set of generators for $A_n$, the alternating group.

The problem is this: Prove that $A_n = \langle (123),(124),\ldots,(12n)\rangle$. I had cogitated this problem for quite awhile, and haven't been able to come up with anything. The only good idea (...
2
votes
3answers
65 views

Which inverse multiplicative groups modulo $n$ are cyclic or not

I've found nothing about this in my book neither in the internet. Also the wikipedia article about inverse multiplicative modulo $n$ is poor. So, I need prove that $$\mathbb Z_n^*$$ is cyclic for $...
5
votes
4answers
129 views

Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi :...
0
votes
2answers
65 views

Exercise about finding group isomorphisms

So, i've just learned about groups homomorphisms and isomorphisms. I know that an homormorphism betweet two groups is a function such that $$\phi(a\square b) = \phi(a)\star \phi(b)$$ And when the ...
3
votes
1answer
82 views

All primes $p$ for which $x^{2} +1$ irreducible over $\mathbb{F}_{p}$

Let $\mathbb{F}_{p}$ be the field with $p$ elements. Determine all primes $p$ for which $x^{2}+1$ is irreducible over $\mathbb{F}_{p}$. Here's what I've got. I think this might be finished but I'm ...
1
vote
2answers
76 views

Subgroup of a group with 24 elements.

Suppose that $G$ is a finite group of order $24$, which has four $3$-sylow subgroups. We know that may contain $1$ or $3$ 2-sylow subgroup. How can I prove that there only exists one $2$-sylow ...
4
votes
2answers
158 views

Consequence of First Homomorphism Theorem?

Let $\phi:G\to\bar G$ be a surjective group homomorphism with kernel $N$. Then the first homomorphism theorem tells us that $G/N\cong\bar G$. My question is this: Lagrange's theorem also tells us ...
4
votes
1answer
185 views

A question on subgroups of a finite group

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$. Prove that $K$ is a subgroup of $H$. So far we found that $o(K)$ divides $o(H)$...
1
vote
1answer
22 views

Questions about terminology (transpositions)

A cycle with only two elements is called a transposition. For example, the permutation of $\{1, 2, 3, 4\}$ that sends $1$ to $1$, $2$ to $4$, $3$ to $3$ and $4$ to $2$ is a transposition (specifically,...
3
votes
2answers
53 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help would ...