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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3
votes
2answers
17 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
8 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
49 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
25 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
votes
2answers
62 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.
1
vote
0answers
20 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 ...
1
vote
2answers
39 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
votes
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
votes
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 [on hold]

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
vote
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
votes
1answer
36 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
vote
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 $?
-2
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
votes
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
votes
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
vote
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
vote
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
88 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 ...
0
votes
1answer
55 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
54 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 ...