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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An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
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83 views

why does table look like this? - multiplicative table for Group

I have a Group $\mathbb{Z_2}[x]/q$ where $q(x)=x^3+x+1$ In my textbook, there a multiplication table: but I dont understand how the last $x+1$ comes. it should be $(x^2+x+1)*(x^2+x+1) \quad mod ...
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102 views

Subgroups of Cyclic $p$-Groups

Theorem Let $G$ be a cyclic group of order $n$, generated by $x$. Then every subgroup of $G$ is cyclic. The proof is as follows: let $H$ be a proper non-trivial subgroup. Then there exists a ...
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If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $<\chi_N , \psi>_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$.

If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $\langle\chi_N , \psi\rangle_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$. Can anybody help me to prove that?
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On the Example of J. Alperin

In the paper "Large Abelian Subgroups of $p$-Groups" by J. Alperin, the author constructs an example of a $p$-group of order $p^{3n+2}$ ($p>2$) in which any abelian subgroup has order at most ...
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Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
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A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
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134 views

Is there a simple way to show that wreath product is associative?

Is there a simple way to show that wreath product is associative? If your proof is short, please write it explicitly. Thank you.
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85 views

Equality in commutator subgroup

I will start by apologizing if this questions seems twisted. I am reading the paper Cohomology theory of groups with a single defining relation (Lyndon, $1950$) and the question comes from page $659$ ...
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52 views

Subgroups of $G^n$

Let $G$ be a non-Abelian group and let $n$ be a (large) positive integer (eventually going to infinity). For simplicity, let's take $G=\mathbb{D}_6$ (Dihedral group of order $6$). Is there a way to ...
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107 views

Commutators in finite groups

I get stuck with this question : Let $g$ be an element in a finite group $G$, and let $k$ be an integer coprime to $|g|$. Prove that $g$ is a commutator in $G$ if and only if $g^k$ is a commutator. ...
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164 views

The first cohomology of group

I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
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136 views

Conjugacy classes in $SL(n,q)$

If $q$ is a prime power, then the conjugacy classes in $GL(n,q)$ are determined completely by minimal polynomials. Now in the subgroup $SL(n,q)$, these conjugacy classes can split, in the sense they ...
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232 views

On Sylow subgroup of simple group PSL(2,p)

Let $G$ be finite group such that $Z(G)=1$ and the number of Sylow $p$-subgroups of $G$ equal to the number of Sylow $p-$subgroups of $PSL(2,r)$ for every prime $p$ ( $r$ is prime and $r^2$ not ...
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166 views

About a Sylow subgroup of a product

Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of G such that $P=(P\cap H)(P\cap K)$. I looked at the proof here, but I ...
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137 views

Terminology in Dickson's book

This appears in Dickson's "Linear Groups with an exposition of the Galois field theory", page 50, chapter 4. My question is: how would the above "translate" in modern terms? In particular, do we ...
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140 views

The class of groups all of whose subgroups have a specific property

For a subgroup $H$ of the finite group $G$, define $C(H)$ to be the set of all subgroups of $G$ which are permutable with $H$, i.e. $C(H) \:= \{K \leq G: HK=KH \}$. My question is: can the class of ...
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200 views

A subclass of metacyclic groups

A group $G$ is metacyclic if it contains a cyclic normal subgroup $N$, such that $G/N$ is also cyclic. $N$ is called the kernel, and the order of $G/N$ is the index of the metacyclic group. There is ...
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146 views

A class of finite groups

Let $G$ be a finite group having the property that for any prime $p$ dividing $|G|$, it has a subgroup H with $[G:H]=p$. What can be said about these groups? I believe I can prove that they must be ...
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Blocks and covering modules

I have a question relating to p109 of Local representation theory by JL Alperin. Let $G$ be a finite group and let $N$ be a normal subgroup. If $B$ is a block of $G$, why must $B$ be a summand of ...
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232 views

[H,K] abelian if K centralizes [H,K]

Here is a paraphrased version of problem 4B.4 in Isaacs's Finite Group Theory: Let $G$ be a group and $X,Y$ subgroups of $G$, such that $Y$ centralizes $[X,Y]$. If $X$ is normal in $G$, show ...
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Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
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+50

Proof of the Frobenius Schur indicator

I am trying to prove the Frobenius-Schur indicator for $\chi$ irreducible character. \begin{equation} i_{\chi} = \begin{cases} 0, & \text{if $\chi$ is not real valued} \\ \pm1, & ...
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Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
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28 views

If $C_A(F(A)) \le F(A)$ and $C_B(F(B)) \le F(B)$, then this also holds for $AB$ if $A,B \unlhd G$.

Let $A, B \unlhd G$ be normal subgroups of a finite group $G$ such that $$ C_A(F(A)) \le F(A) ~\mbox{ and }~ C_B(F(B)) \le F(B). $$ where $F(G)$ denotes the Fitting Subgroup of $G$. I want to show ...
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38 views

does minimality condition imply normal p-sylow subgroup >

Assume that $G$ is a finite group, and $p$ is a prime number dividing the order of $G$. Let $\cal C=\cal C(H)$ be the following condition : "$H$ is a normal subgroup of $G$ and $|G/H|$ is coprime to ...
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Relations between the commutator of two subsets and set theoretical notions

If $X,Y$ are two subsets of some group $G$, then $$ [X,Y] := \langle [x,y] : x \in X, y \in Y \rangle $$ is the commutator subgroup generated by $X$ and $Y$ (where $[x,y] := x^{-1}y^{-1}xy)$. Are ...
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Matrix Elements of Real Represententations

Suppose that $G$ is a finite group and we have a unitary irreducible representation $\rho:G\rightarrow \hom V$. Suppose we fix a basis $\{e_i\}_{i\geq 1}$ of $V$ and with respect to this basis we have ...
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Group presentation: How can we determine the group with the presentation below?

Please how can we determine the finite group whose presentation is given as: $$G:=\left\langle g,h|g^4=h^4=1,hg=g^{-1}h \right\rangle?$$
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$gcd(|G|, |Aut(G)|)=1$ means G is abelian?

Prove the following assuming that G is finite group with $gcd(|G|, |Aut(G)|)=1$ a)G is abelian (done) b) Every Sylow subgroup of G is cyclic of prime order. G is abelian than every sylow unique, ...
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a finite group of order $360$ never has an irreducible character of degree $15$

The following problem is an exercise in character theory: "Prove that a finite group of order $360$ never has an irreducible character of degree $15$." Could you help me for it?
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Non-isomoprhic semidirect products and their centers

I have a group $G \cong P \rtimes \mathbb{Z}_4$, where $|P| = p^2$ ($p$ is an odd prime). Do I have enough information to determine whether the two possible structures of $P$ both yield unique ...
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Adding relators and normal closure

I learned the notion of group presentation. By definition, a group described by a presentation is the quotient of some free group by the normal closure of the relators. In general, given a group ...
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51 views

Sylow 5-subgroups of groups of order $2^n5^m$ are normal

My textbook says: Show that a group of order $2^n5^m, m, n \ge 1$ has a normal 5-Sylow subgroup. I've been banging my head against this problem for days with no success, how can I prove this?
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69 views

Capable group of order 32

A group that can be written as $G/Z(G)$ for some group $G$ is called capable. Can someone list the capable groups of order 32?
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A group of order $2^67$

Let $G$ be a finite group of order $2^6\times 7$. Also $G$ has at least $5$ irreducible characters of degree $7$. I want to prove that with this condition there exists no character of even degree in ...
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What is the most mundane & intuitive meaning of the radical of a finite group?

I am asking this question because I am trying to solve this problem from a class note: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G)).$ The note's ...
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102 views

Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
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Over which fields is a $G$-module reducible?

Let $K$ be a field of characteristic zero, or if this is too general, an algebraic number field. Let $G$ be a finite group and $V$ an irreducible and finite-dimensional $KG$-module. Let $\chi$ be the ...
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Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
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Characterize all $\mathbb Z[i]$ modules of order $21$ and $65$

I am trying to figure out how many $\mathbb Z[i]$ modules (up to isomorphisms) are of $21$ and $65$ elements. I've done a few similar exercises for finite abelian groups ($\mathbb Z$ modules) but I am ...
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63 views

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ?

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ? ( By Cauchy's theorem I can show that there are elements of order $2$ and $3$ but cant proceed ...
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Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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A Question on the Quotient Group and/or set of cosets

I'm just confused about a somewhat simple fact about quotient groups. If we have: $$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we ...
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If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
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Possible number of Groups order N

For when is the number of groups of some order n more than n? For example, let say this happens at $n=3$ then that would mean that there are more groups of order 3 than 3.
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How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
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68 views

Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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Parametrizing a group element

I'm looking at the problem of parametrizing a group element $g \in SL(n,\mathbb{R}).$ I think I understand the concept of parametrization $-$ just a change of coordinates, right? But I don't ...
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Group's morphisms

I know that the constant map equal to $1$ and the signature are two group's morphisms from the group of permutations $(\mathcal S_n,\circ)$ to the group $(\Bbb C^*,\times)$. My question is: Can we ...