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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Factorizations of Finite Abelian Groups

Every finite abelian group $G$ can be uniquely written as $$\mathbb{Z}/{d_1\mathbb{Z}} \times \mathbb{Z}/{d_2\mathbb{Z}} \times \cdots \times \mathbb{Z}/{d_r\mathbb{Z}},$$ where $d_i$ divides ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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 $M_{12}$ ...
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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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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 $x$. ...
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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 ...
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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 $k=\lfloor \log_2 (n-1)\rfloor$.
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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 ...
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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 ...
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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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1answer
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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 ...
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2answers
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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 ...
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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: ...
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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.
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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, ...
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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 $n^2$ ...
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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 ...
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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 ...
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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 ...
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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 $G$.
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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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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 ...
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2answers
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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 ...
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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, ...
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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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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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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 ...
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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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1answer
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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 ...
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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.
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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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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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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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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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Hensel lifting when not a power of a prime

Say you have the equation $x^2 + x + 47 = 0$ and that you want to determine the solutions in $\mathbb{Z}/1715 \mathbb{Z}$. Note that $1715 = 7^3 \cdot 5$. Then, using Hensel's lemma, one can find the ...
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Extra-Special $p$ group and complement

Let $G$ be an extra special $p$ group of order $p^{2n+1}$, $n\geq 2$. Does $[G,G]$ necessarily have a complement in $G$? I dont think so, but I am not sure. Sorry this should be a very silly ...
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Number of groups of a certain order given: 1) finitely generated abelian, 2) subgroups, 3) not necessarily finitely generated

1) How many finite abelian groups are there of order $1000$? Well via the fundemental theorem of finitely generated abelian groups, we look at the factorisation for $1000$. $1000=2^3*5^3$ and there ...
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1answer
30 views

Subnormal series and indices

I'm trying to solve this problem: Let be $H, K$ subgroups of a finite group $G$. Suppose that exists one serie of subgroups such that $G=G_{0}\triangleright G_{1} \triangleright \ldots ...
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1answer
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transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
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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 $H\subset K\subset G$? ...
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1answer
23 views

can i do this transformation with any finite group?

I have a finite alphabet $\{e_1, e_2, \cdot \cdot\cdot, e_N, a_1, a_2, \cdot\cdot\cdot, a_n \}$ where we pair $e_i$ and $a_i$ as ``opposites'' - like opposite vertexes on a regular $2N$ sided polygon ...
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200 views

Number of elements in a finite abelian groups

Is the following true? Let $G$ be a finite abelian group with a minimal generating set $S$. By minimal generating set I mean we cannot reduce the cardinality further. Let $S=\{a_1,a_2,\ldots ,a_k\}$ ...