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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1
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2answers
30 views

Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
3
votes
2answers
33 views

Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
4
votes
2answers
30 views

About the elements of a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$

Let $G$ be a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$. I would like to prove that for every $g \in G$, $\det(g) \in \lbrace -1,1 \rbrace$. Here are my ideas : since $G$ is a finite subgroup ...
0
votes
0answers
18 views

Calculation of Fixed point set

Let $p,q$ be two distinct primes and let $G_p$ and $G_q$ be nontrivial $p$-group and $q$-group respectively. Let $G = G_p \rtimes G_q$. Embed $G_q \subset U(n_1), G \subset U(n) $ where $U(n_1), ...
1
vote
2answers
53 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
2
votes
1answer
26 views

Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...
0
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2answers
64 views

Automorphism of $D_6$

I need find an explicit way to express the group $Aut(D_6)$, and I have not idea how write this group, maybe this is an semidirec product of some groups but I don´t see this. thanks.
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0answers
21 views

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
2
votes
2answers
40 views

Generators in group $Z^*_{p}$

show that $g=2$ is a generator of group $Z^*_{19}$ Can anyone explain me how i can show in this example and generally that an element is a generator in a group?
2
votes
1answer
36 views

$G$ be a group of order $p^n$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$?

Let $G$ be a group of order $p^n$ , where $p$ is a prime and $n \in\mathbb N$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$ ?
2
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1answer
36 views

On the number of Sylow subgroups in Symmetric Group

If $G$ is a finite group, and $P$ is a Sylow-$p$ subgroup of $G$, then the number of Sylow-$p$ subgroups in $G$ is at most $|G|/|P|$. In the Symmetric group $S_n$, the bound is attained only for ...
0
votes
0answers
11 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
103 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
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1answer
31 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
62 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
48 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 ...
1
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0answers
39 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
33 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
26 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
27 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 ...
0
votes
0answers
29 views

A question about primitive idempotent of group algebra [closed]

How to prove: $e_j$ is primitive idempotent of group algebra $\cal{L}$ iff $\forall\ t\in \cal{L}\ $, $e_j^2=e_j$ and $e_j t e_j=\lambda_te_j$. Or in which book can I find the proof.
4
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1answer
37 views

Cubic Planar Graphs have $2^m-1$ Hamilton Cycles

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
31 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
36 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
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4answers
48 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 ...
0
votes
0answers
33 views

Let $ G $ is finite nilpotent group. can we say $ G $ is $ p $-supersoluble for any prime $ p $?

Let $ G $ is finite nilpotent group. Thus $ G $ is $ p $-nilpotent group for any prime $ p $. Now can we say $ G $ is $ p $-supersoluble for any prime $ p $?
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votes
0answers
19 views

Existence of groups with cyclic 2 Sylow subroups S o(S) = $2^k$

Given a number $2^k$, is there a finite group G whose 2 Sylow group is of order $2^k$ and is cyclic? The interest in this is that such groups have a characterization which applies in particular to ...
0
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0answers
15 views

finite subgroups of O(3,1) [closed]

What are the finite subgroups of $O(3,1)$ or $O(p,q)$ more generally? Apparently "this question body does not meet our quality standards."
0
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0answers
57 views

The trivial group, multiplication, 0 and 1 [closed]

The trivial group has one element, call it $\{e\}$. Use the binary operator for multiplication, $*$. Setting $e=1$ will work since: the group has closure: $1 * 1 = 1$ the identity element is $1$ the ...
0
votes
1answer
35 views

Can one modify the generators of a transitive group to get an intransitive group while preserving conjugacy classes?

There is a general question I'm interested in: given $g$ and $h$ with $H=\langle g,h \rangle$ a transitive subgroup of $S_n$, when is it possible to find $g',h'$ so that $H'=\langle g',h' \rangle$ is ...
1
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2answers
43 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
52 views

Is there a simple proof of Frobenius's theorem using Sylow theory? [closed]

I ask because the proof I always stumble upon uses double induction and properties of the totient function. The statement of the theorem is: If $n$ divides the order of a finite group $G$, then ...
2
votes
0answers
12 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 ...
0
votes
0answers
20 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
24 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
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0answers
22 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
29 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 ...
6
votes
1answer
72 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 ...
0
votes
0answers
37 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
90 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
21 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 ...
2
votes
1answer
24 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 ...
3
votes
4answers
161 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 ...
1
vote
1answer
66 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
22 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 ...
4
votes
5answers
79 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 ...
1
vote
1answer
26 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
41 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
52 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 ...