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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3answers
111 views

To prove there is only a finite number of subgroups in G

if $H$ is subgroup of finite index in $G$. prove that there is only a finite number of distinct subgroups in $G$ of form $aHa^{-1}$.
2
votes
0answers
47 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,3\}$ for $g \ne 1$, and stabilizers are t.i. subgroups, then the Sylow $3$-subgroups have maximal class

Let $G$ be a transitive permutation group such that every nontrivial element fixing some point fixes exactly $3$ points. Also assume that for $g \notin N_G(G_{\alpha})$ we have $$ G_{\alpha} \cap ...
1
vote
0answers
45 views

If $S = A \times B$ with $|A| \ne |B|$ and both cyclic, and $S \in \mbox{Syl}_3(G)$, then $G$ has a normal $3$-complement

Let $G$ be a group of odd order, and let $S$ be a Sylow $3$-subgroup of $G$. Assume that $S$ is abelian with $S = A \times B$ and $A,B$ cyclic. If $|A| \ne |B|$, then from Burnside's transfer theorem ...
5
votes
1answer
62 views

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
1
vote
2answers
34 views

A certain centralizer group in finite group of order $3^5$ of maximal class contains every normal subgroup of index greater than $3$

Let $G$ be a finite group of order $3^5$ and maximal class, i.e. if $K_{i+1}(G) = [K_i(G), G]$ denotes the subgroups of the lower central series, we have $K_5(G) = 1$ and $K_4(G) = Z(G)$. Set $G_1 = ...
3
votes
0answers
63 views

Let $p$ be a Mersenne prime and $G$ an extension of $C_p$, then we have a subgroup in which $C_p$ has a complement

Let $p$ be an Mersenne prime, i.e. $p + 1$ is a power of $2$. Suppose $G$ is an extension of the cyclic group $C_p$ of order $p$ by a group $\overline H$ isomorphic to $PSL(2, p+1)$ or the Suzuki ...
0
votes
0answers
26 views

$g_1,g_2 \in G$ such that for any complex character $\chi$ , $\chi (g_1)=\chi(g_2)$ ; does $g_1,g_2$ belong to same conjugacy class?

We know that any character on a finite group is a class function i.e. they each take a constant value on a given conjugacy class . Is the converse true ? that is let $G$ be a finite group , $g_1,g_2 ...
1
vote
1answer
32 views

When Center of group is a subset of Normalizer($Z(G) \subset N(a)$).

DEFINITION: If $a \in G$, then $N(a)$, the normalizer of $a$ in $G$, is the set $N (a) = \{ x \in G | xa = ax \} $. $Z(G)$ is the center of the group. I found the following proof - Lemma: If ...
1
vote
1answer
30 views

Why all $3$-cycles are conjugate in $\mathfrak S_5$?

Why all $3$-cycles are conjugate in $\mathfrak S_5$ ? Is it the case for all $n$ or only on $\mathfrak S_n$ ? I mean, in $\mathfrak S_n$ all $3$ cycle are conjugate or not ? (it doesn't look to be the ...
3
votes
2answers
104 views

How many groups of order $2500$ are there?

I aborted the GAP-calculation of $Size(ConstructAllGroups(2500))$ after about $3$ hours. $gnu(2500)$ seems to be a very hard case. Does anyone know $gnu(2500)$ (The number of groups of order ...
2
votes
2answers
55 views

Known result about such $p$ groups

Let $G$ be a finite $p$ groups with, $|G'|=p$ $|G:Z(G)|=p^2$ $Z(G)$ is cyclic. $1)$ Can $G$ have nonabelian maximal subgroup ? It is clear that all maximals containing the center are abelian. Is ...
6
votes
0answers
86 views

Groups whose quotients are cyclic

It is well known that a finite group whose all proper subgroups are cyclic is either cyclic or direct product of quaternion group with cyclic group of odd order (am I correct?) Question: What are the ...
0
votes
1answer
46 views

Can a Simple Group possess this property? [closed]

If a simple group G is of order 168 then can I find subgroup of order 7 of G ? If so, then what is the number of subgroups of G of order 7 ?
3
votes
2answers
55 views

Show that if $G$ is a finite group of even order, there must exist an element $a\in G, a\ne e$ s.t. $a^2=e$ [duplicate]

As order of group is even then $G$ can contain an element of order 2 as 2|even but how it is must?
2
votes
1answer
34 views

Find the minimal polynomial over a field

I have two similar questions: 1). Find the minimal polynomial for $a^{-1}$ (a to the power of minus 1) over $F_3$. $a$ is the root of the polynomial $x^3-x+1$ in $F_3[X]$ I have used the division ...
2
votes
0answers
40 views

Rigidity of conjugacy classes of finite centerless groups

Let $G$ be a finite group and let $C_1,...,C_k$ be conjugacy classes of $G$. We define the following set: $\Sigma$ = $\{(g_1,...,g_k) \in \prod_{i=1}^{k}C_i \hspace{1mm} | \hspace{1mm} ...
0
votes
3answers
47 views

If $G$ is a groupe such that $|G|=p^m k$, does $G$ has a subgroup of order $p^n$ with $n<m$.

If $G$ is a groupe such that $|G|=p^m k$, does $G$ has a subgroup $H$ s.t. $|H|=p^n$ with $n<m$ ? I know that $G$ has a $p-$sylow subgroup, i.e. a group of order $p^m$. I also know that $G$ has ...
1
vote
1answer
32 views

There is an $m$ such that $M^{m}-I_{n}$ is not invertible for all $M \in GL({n,q})$

We have a general linear group over a finite field. I need to show that for every $M$ in my group I can find an integer $m$ such that $$M^{m}-I_{n}$$ is not invertible. I know this happens because of ...
2
votes
1answer
45 views

When is the centralizer of a subgroup equal to the center?

Let $G$ be a group, and $H\leq G$ be a subgroup. When is $C_G (H)=Z(G)$? Similar to this question, which is about the centralizer of an element rather than of a subgroup: When is the centralizer and ...
3
votes
2answers
49 views

Modules over associative algebras are just special cases of “ordinary” modules over rings?

By module over a ring, I mean always a right-module. All rings are supposed to be unital, and the module fulfills $m\cdot 1 = m$. If $R$ is commutative and $M$ a right-module, we can define $rx := xr$ ...
3
votes
2answers
102 views

How can I calculate $gnu(17^3\times 2)=gnu(9826)$ with GAP?

I tried to calculate the number of groups of order $17^3\times 2=9826$ with GAP. Neither the NrSmallGroups-Command nor the ConstructAllGroups-Command work with GAP. The latter one because of the ...
2
votes
0answers
58 views

Is there an efficient method to decide whether $gnu(n)<n$ , $gnu(n)=n$ or $gnu(n)>n$?

Denote : $gnu(n)$ = number of groups of order $n$ It is much easier to decide whether a natural number $n$ is group-deficient ($gnu(n)<n$) , group-perfect ($gnu(n)=n$) or group-abundant ...
1
vote
1answer
15 views

Invariant factors and elementary divisors of an abelian group

I have to find the elementary divisors and invariant factors of : $$ \mathbb Z_6\oplus\mathbb Z_{20}\oplus\mathbb Z_{36}$$ I'm following this. I think that elementary divisors are ...
0
votes
0answers
31 views

non-commutative association scheme [duplicate]

Which group can be work on next the example makes a non-commutative association scheme Ex: Let a group $G$ act transitively on a finite set $X$. Then $G$ acts naturally on the set $X×X$ by ...
4
votes
1answer
63 views

Product of two abelian subgroups

A theorem of Ito says that if $G=AB$ where $A,B$ are abelian subgroups of $G$ then $G'$ is abelian. It was an exercise in a book, to prove, without using above fact, that If $G$ is finite group ...
2
votes
1answer
55 views

The equation $|A^{-1}A|=|A|^2-|A|+1$ for finite subsets of a group

Let $A$ be a finite subset of a group $G$. It is clear that $|A^{-1}A|\leq|A|^2-|A|+1$. If $A$ is singleton or $A=\{1,a\}$ with $O(a)\neq 2$ then the equality holds. Now, can somebody characterize all ...
4
votes
2answers
89 views

$S_5$ does not act transitively on $\{1,2,3\}$?

I have seen a statement: let $S_n$ act transitively on a set with $m$ elements. Then, $m\leq 2$ or $n \leq m$. I was able to prove it, so I believe it. However, I find this strikingly unintuitive. For ...
8
votes
1answer
83 views

The smallest symmetric group $S_m$ into which a given dihedral group $D_{2n}$ embeds

Several questions, both here and on MathOverflow, address the issue of determining for a given group $G$ the smallest integer $\mu(G)$ for which there is an embedding (injective homomorphism) $G ...
4
votes
3answers
55 views

Group and subgroup

I just learned that a group is a set together with an operation such as $<G,•>$. And a subgroup is a subset H of G, which forms a group under "•". I got confused by the definitions because I ...
0
votes
1answer
40 views

show the form of a subgroup when the index and order are relatively prime

Let $G$ be a finite group and $K$ be a normal subgroup of $G$. If $\gcd([G:K],|K|)=1$, then $K=\{x^i | x \in G\}$, where $i=[G:K]$. I know that if we consider $G/K$, since the index is $i$, $g^i ...
1
vote
0answers
20 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
6
votes
1answer
191 views

Is $gnu(2304)$ known?

I wonder whether the number of groups of order $2304=2^8\times 3^2$ is known. GAP exited because of the memory. $gnu(2304)$ must be greater than $1,000,000$ because of $gnu(768)=1,090,235$ and ...
0
votes
1answer
27 views

The central product of two cyclic subgroups of prime power order for one $p$ is isomorphic to direct product of two cyclic groups of prime power order

For two groups $G_1, G_2$ and two central subgroups $U_1 \le Z(G_1), U_2 \le Z(G_2)$ which are isomorphic by some given $\mu : U_1 \to U_2$ the central product is the group $$ (G_1 \times G_2) / D $$ ...
3
votes
1answer
35 views

Automorphism of cyclic $p$-group

A cyclic group of order $p^n$, $n\geq 1$, always has an automorphism of order $p-1$ (well known). Let $C_{p^n}=\langle x\colon x^{p^n}=1\rangle$, and $H=\langle x^p\rangle$, the unique subgroup of ...
7
votes
2answers
84 views

Making a proof precise of “Aut$(Q_8)\cong S_4$”

I know, with some machinery, how to prove that Aut$(Q_8)\cong S_4$. My question here is about not how to prove, but is about an incomplete proof (I feel) given by a student to me (it should be ...
8
votes
3answers
191 views

How many groups of order $2058$ are there?

I tried to calculate the number of groups of order $2058=2\times3\times 7^3$ and aborted after more than an hour. I used the (apparently slow) function $ConstructAllGroups$ because $NrSmallGroups$ did ...
0
votes
1answer
27 views

The Frattini subgroup in $p$-groups and factor groups

Let $G$ be a finite group, then its Frattini subgroup $\Phi(G)$ is the intersection of all maximal subgroups of $G$. If $N \unlhd G$, then in general we have $$ \Phi(G)N / N \le \phi(G / N) $$ as ...
1
vote
0answers
49 views

A GAP code for maximum and minimum cardinals of some classes of subsets of a finite group

Let $G$ be a finite group with a fixed subset $A$. Put $$ S_r(A)=\{B\subseteq G : |AB|=|A||B|\; \& \; B \; \mbox{is inclusion-maximal with respect to this property}\} $$ $$ M_r(A)=\max\{|B| : ...
0
votes
2answers
32 views

How many disjoint product of two $2$-cylces are there in $S_5$ ?

How many disjoint product of two $2$-cylces are there in $S_5$ ? In general I'm having trouble in determining no. of disjoint product of cylces . Please help . Thanks in advance
2
votes
1answer
43 views

show that a Sylow p-group lies in the center of $G$

I'm stuck with the following problem: Let $G$ be a finite group and $p$ be the smallest prime dividing $|G|$. Suppose the Sylow p-subgroup $H$ of $G$ is normal and cyclic. Show that $H$ lies in the ...
0
votes
1answer
30 views

Number of subgroups of elementary abelian group not contained in one factor.

Let $B = A \times \langle x \rangle$ be elementary abelian of order $p^n$. Let $D$ be a subgroup of order $p^k$ with $k \le n-1$. Then we have $$ m \cdot |A : D \cap A| $$ different subgroups with ...
1
vote
0answers
28 views

Simple random walk on the $N$-cycle

I am considering the following example: In my lecture notes we noted that "the functions $(\phi_j)_j$ form a basis". I think they refer to the space $\mathbb{C}^G$ where $G$ is the above ...
9
votes
4answers
107 views

Why are there $12$ automorphisms of $\Bbb Z\oplus \Bbb Z_{3}$?

Let $A:=\Bbb Z\oplus \Bbb Z_{3}$, then what is $|\text{Aut}(A)|$? My answer is $4$ but the correct answer (without explanation) turns out to be $12$! How come? Well my understanding is, it just ...
4
votes
0answers
50 views

A “new” formula relating the quotients of the upper central series, method of proof and background information

For a finite group $G$ the upper central series is defined inductively by $$ Z_1(G) := Z(G), \qquad Z_{i+1}(G) / Z_i(G) = Z( G / Z_i(G) ). $$ Now I am interested in generalising this formula, i.e. ...
0
votes
1answer
29 views

Projective space and its basis

I am trying to solve an exercise from the book "Permutation Groups" by J. Dixon and B. Mortimer. Later, I asked a similar question about the basis of Affine geometry " Affine geometry and its ...
2
votes
0answers
34 views

Faithful irreducible representation of a finite $p$-group

I want to solve the following exercise in the representation theory field: A finite $p$-group $G$ has a faithful irreducible representation over an algebraically closed field whose characteristic is ...
3
votes
1answer
49 views

About an irreducible representation over an algebraically closed field

I want to prove the following statement that is an of the book "A course in the theory of groups" by D. Robinson: Let $n$ be the degree of an irreducible representation of a finite group $G$ over an ...
2
votes
2answers
82 views

A finite abelian group $A$ is cyclic iff for each $n \in \Bbb{N}$, $\#\{a \in A : na = 0\}\le n$

Let $A$ be a finite abelian group. Prove that $A$ is cyclic iff for each $n \in \Bbb{N}$ $$\#\{a \in A : na = 0\}\le n.$$ Any help or hint will be appreciated.
1
vote
4answers
50 views

Subgroup with index equal to smallest prime factor normal. How can I prove this?

Let $G$ be a group of order $n>1$ and $p$ the smallest prime factor of $n$. Suppose, $H$ is a subgroup of $G$ and $[G:H]=p$. How can I prove that $H$ is normal ? According to Lagrange, ...
4
votes
1answer
57 views

$G$ is abelian when any two non-identity $a$ , $b$ there is an automorphism $\delta$ such that $\delta(a)=b.$

$G$ is a finite group with identity $\mathcal e.$ Suppose for any two non-identity elements $a$ , $b$ of $G$ , there is an automorphism $\delta$ such that $\delta(a)=b.$ Then prove that $G$ is ...