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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0answers
24 views

Permutation of cosets

Let $G$ be a finite group and $\gamma \in Sym(G)$, such that $\gamma (1) = 1$ and $\gamma (gH) = \gamma (g)H$ for all $g\in G$, $H\leq G$. This means $\gamma$ induces a permutation of the left cosets ...
2
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3answers
72 views

Prove that $|GL_n(\mathbb{F})|< q^{n^2}$.

Let $\Bbb F$ be a finite field, say $|\Bbb F|=q$; then we know that $|GL_n(\Bbb F)| < \infty$. But how can we prove that $|GL_n(\mathbb{F})|< q^{n^2}$? I'm guessing because there $n^2$ ...
4
votes
1answer
58 views

Generating series - Finite groups of order $n$

I am wondering if something of interest can be said about one of the two series $$G_1(x)=\sum_{n=1}^{+\infty}{\mathcal{G}(n)z^n}$$ $$G_2(s)=\sum_{n=1}^{+\infty}{\frac{\mathcal{G}(n)}{n^s}}$$ where ...
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0answers
32 views

Are there groups of order $p^4q^2$ which are not semi-direct product?

It is easy to show that if $G$ is a group of order $p^2q^2$, where $p,q$ are primes with correspondings Sylow subgroups $P,Q$, that $G$ is a semi-direct product of $P$ and $Q$. Moreover, if $pq\neq ...
2
votes
2answers
82 views

Classifying groups of order $5 \cdot 11 \cdot 61$

My question is whether I'm classifying groups of order $5 \cdot 11 \cdot 61$ correctly. (This is a qualifying exam question, so I also want to make sure that I'm doing it “efficiently”.) Sylow's ...
22
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1answer
788 views

Rotman's exercise 2.8 “$S_n$ cannot be imbedded in $A_{n+1}$”

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ...
0
votes
1answer
24 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
0
votes
1answer
13 views

Existence of a special Central cyclic subgroup

I have two related questions. Let $G$ be a finite $p$ group. Can we always choose a central subgroup $N$ of order $p$ not contained in the commutator subgroup? Clearly we cannot do that for extra ...
0
votes
1answer
28 views

Abelianization of a $p$ group.

Let $G$ be a $d$ generated finite $p$ group. Let $N$ be normal subgroup of order $p$ contained in $[G,G]\cap Z(G)$. Can we say say that $(G/N)/[G/N,G/N]=(G/N)_{ab}$ is a direct product of $d$ cyclic ...
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0answers
34 views

Covering groups

I am studying the Steiner system $S(5,6,12)$ and the ternary extended Golay code $\mathscr{C}_{12}$. The automorphism group of the Steiner system is the Mathieu group on twelve elements $M_{12}$ ...
2
votes
1answer
58 views

On cyclic decomposition of element in $S_n$

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...
3
votes
1answer
137 views

Which of the following abelian groups are cyclic groups?

Given the abelian groups of order $7425$: $$Z_{33} \times Z_{15} \times Z_{15} , \ Z_{25} \times Z_{297} , \ Z_{45} \times Z_{165} , Z_{55}\times Z_9 \times Z_{15}$$ Which of these groups, if ...
0
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1answer
36 views

Fulton and Harris: Exercise 1.3 in section 1.1

This is exercise 1.3 on page 5 of Fulton and Harris Representation Theory: A First Course. Exercise: Let $G$ be a finite group, let $V$ be an $n$-dimensional $\mathbb C$-vector space and let $\rho: ...
3
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0answers
61 views

Prove that $|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$ [on hold]

Suppose that $G$ is a group of order $n > 1$, prove that $$|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$$ where $k=\lfloor \log_2 (n-1)\rfloor$.
2
votes
1answer
36 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
2
votes
0answers
32 views

Solving the Character table for $A_4$ and derived algebra characteristics.

Say I am given a group with 4 conjugacy classes $C_1, C_2, C_3, C_4$ with orders 1,3,4,4 respectively. I will call the conjugacy class characters $\chi_1,\chi_2,\chi_3,\chi_4$ respectively. I am ...
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0answers
22 views

What's the asymptotic of the radius of the Rubik square Cayley graph?

This post is a sequel of The Rubik Square permutation groups, which should be read first to understand the notation. Question: what's the radius$^*$ of the Cayley graph of $G_n$ generated by the red ...
2
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1answer
67 views

The Rubik Square permutation groups

This post was inspired by this webpage of mathematical challenge due to Mickaël Launay (French). Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as for the following ...
2
votes
1answer
82 views

Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?

I saw this question on an old qualifying exam: Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$? A simpler problem would be to show ...
2
votes
2answers
52 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
5
votes
2answers
77 views

How to obtain a presentation for each group of order $64$

I am new to his forum, and would like to know how to obtain a presentation for each group of order $64$. I wish to do this for all the groups of order $64$. Thanks in advance.
8
votes
1answer
50 views

Upper bounds for the number of intermediate subgroups

Assume that $G$ is a finite group, and $H\le G$ a subgroup of index $n>1$. What can we say about the number of distinct intermediate subgroups $K$, i.e. groups such that $H\subset K\subset G$? ...
3
votes
3answers
72 views

Finding the number of elements of particular order in the symmetric group

I know how to find the order of element in any group $G$, for example the order of $2$ in $\mathbb{Z}_5$ is $5$ as $2 + 2 + 2 + 2 + 2 = [10]_5 = 0 0$, which is the identity in $\mathbb{Z}_5$. But, ...
1
vote
2answers
61 views

Why is $U(10)\not\approx U(12)$?

I'm trying to understand this example on isomorphism but failing to do so. I know that if two groups have a differing number of elements of each order they are not isomorphic. I assume this is what ...
0
votes
0answers
19 views

Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
0
votes
2answers
37 views

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. [duplicate]

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. Prove that, $\exists$ $x\neq e$ such that $x^2=e$ where $e$ represents the identity of $G$.
4
votes
2answers
78 views

Irreducible representations of $\operatorname{GL}_3(\mathbb{F}_q)$

I am trying to find all irreducible representations of $G = \operatorname{GL}_3(\mathbb{F}_q)$. I know that the order of $G$ is $(q^3-1)(q^3 - q)(q^3 - q^2)$ and the number of conjugacy classes is ...
5
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0answers
46 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ ...
2
votes
1answer
36 views

Intuition on Hall subgroups and solvability

I see there are many questions on Hall subgroups, but I can't find one that answers my question. Let $G$ be a finite group. A subgroup $H<G$ is a Hall $\pi$-subgroup if its order is the product of ...
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vote
4answers
365 views

How can I show that $A_5 = \langle (1,2,3), (1,2,3,4,5) \rangle$?

Show $A_5 = \langle (1,2,3), (1,2,3,4,5) \rangle$. Guys how can I show this in the best way.
-1
votes
1answer
36 views

If $G$ is isomorphic to $S_n$, does there exist a subgroup $H$ in G such that H is isomorphic to $A_n$? [closed]

By Cayley's Theorem, every group is isomorphic to a permutation group. Then suppose $G$ is isomorphic to $S_n$, then is it true that there exists a subgroup $H$ in $G$ such that $H$ is isomorphic to ...
3
votes
0answers
44 views

Schur multiplier of “large” groups.

Let $G$ be a finite group and let $M(G)=H^2(G,\mathbb{C}^*)$ be its Schur multiplier. For "small" groups I can compute the Schur multiplier by hand in terms of corresponding roots of unity. However, ...
2
votes
3answers
1k views

Possible Class equation for a group

Determine the possible class equation for a group of order 21? Until now I have found the following: $1+3+3+7+7$ $1+1+1+3+3+3+9$ $1+1+1+1+1+1+1+7+7$ $1+1+1+1+1+1+1+1+1+3+3+3+3$ $1+1+1+\cdots +1 ...
2
votes
1answer
56 views

What does $(G:G')$ mean?

I'm trying to teach myself some ring theory from a book, and have come across this sentence: "There are $(G:G') = 4$ linear characters" where $G$ is a group, and $G'$ is the derived group. I ...
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0answers
37 views

Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
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0answers
34 views

Number of complements

If $G$ has a normal Hall subgroup $U$ then $U$ has a complement $V$ in $G$ and all of these complements are conjugate. Can we say something about the number of complements? Or in other words: How ...
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votes
0answers
31 views

On Thompson conjecture

Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Consider $N(G)=N(A_n)$ and $n/2<p,q<n$, where $p,q$ are prime number. Moreover, assume that Sylow $p$-subgroups and Sylow ...
2
votes
1answer
50 views

Subgroup of symmetric group Sn

Suppose $G$ is a transitive subgroup of $S_n$ such that it there exist $\sigma, \tau \in G$ such that $\sigma$ is an $n-1$-cycle and $\tau$ is a transposition. Prove that $G = S_n$ I just don't ...
4
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1answer
54 views

How to compute the automorphism group of split metacyclic groups?

I am trying to calculate the automorphism group of an affine subgroup $$G=\mathbb{Z}_p\rtimes\mathbb{Z}_{k}\leq\text{AGL}(1,p).$$ One might guess $\text{Aut}(G)=\text{AGL}(1,p)$. And this matches ...
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vote
2answers
44 views

On conjugacy class size of an element and its order.

Let $G$ be a finite group and $x\in G$. Also we denote the conjugacy class of $x$ in $G$ by $x^G$. I want to know if there is any relation between $|x|$ and $|x^G|$? Suggestions would be appreciated.
3
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1answer
49 views

Prove the group is a direct product [closed]

Let $G$ be an abelian group of finite order $n = mk$ with gcd$(m,k) = 1$. For $r=m,k$, let $G(r) = \{g \in G: g^r = 1 \}$ . Prove that $G = G(m) \times G(k)$.
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votes
0answers
23 views

Subgroups of finite reflection groups

I am trying to understand finite reflection groups. Given a connected finite reflection group generated by $m$ reflections and let $S$ be a set of simple roots. Let $I \subset S$ be a subset of the ...
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1answer
23 views

Finding cosets of a quotient group: List the cosets of $HN/N$

In the group $\Bbb Z_{24}$, let $H=\langle 4\rangle $ and $N=\langle 6\rangle $ 1) List the elements of $HN$. I found $HN=\{0,2,4,\cdots,22\}=\langle 2\rangle$ 2) List the elements of $H\cap N$. I ...
2
votes
0answers
22 views

How many symmetries does the Cauchy-Schwarz inequality have?

The symmetries of the Cauchy-Schwarz inequality $(x_1^2 + \cdots + x_n^2)(y_1^2 + \cdots + y_n^2) \ge (x_1y_1 + \cdots + x_ny_n)$, as a subgroup of the symmetric group on the $2n$ letters $x_1,\ldots, ...
1
vote
3answers
56 views

Is $H\cup K$ a group?

If $H$ and $K$ are subgroups of $G$ is $H\cup K$ also a subgroup of $G$? We have identity for sure(since it is in $H$ or $K$), associativity is absorbed. Thus we only need to see if inverses and ...
7
votes
1answer
87 views

What's the smallest non-$p$ group that isn't a semidirect product?

Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups. For example, a ...
27
votes
1answer
2k views

Six Frogs - Puzzle (order reversal using special transpositions)

I had come across a puzzle: The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its ...
2
votes
0answers
50 views

Short exact sequence of groups

Let $1\to N\xrightarrow{\text{i}} G\to H\to 1$ be a short exact sequence of groups. Let $f:G\to N$ be a group homomorphism such that $fi: N\to N$ is an isomorphism. What are some "good" things we can ...
2
votes
0answers
327 views

Find order of given factor group

I'm trying to find the order of this factor group: $$(\mathbb Z_{12}\times\mathbb Z_{18}) / \langle (4,3)\rangle.$$ The order of the factor group is just the number of elements in it (aka the ...
0
votes
1answer
26 views

Soluble(solvable) and nilpotent groups

Defn 1.1. Let $\gamma _{0}(G)=G$, and $\gamma _{c}(G)=[\gamma _{c-1}(G),G]$ for $c\geq 1$. The lower central series of $G$ is a chain of subgroups of $G$: $$G=\gamma_0(G) \geq \gamma_1(G) \geq \cdots ...