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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On automorphisms of finite abelian group

Let $G$ be a finite abelian group such that $a, b\in G$ and $\mid a\mid=\mid b\mid$. Then does there exist an automorphism of $G$ such that $\alpha(a)=(b)$? Thank you
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Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
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Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
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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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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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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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139 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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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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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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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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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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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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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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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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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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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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[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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Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
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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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25 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, ...
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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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Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
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Need help in understanding the question. Elemntary number theory

I have this question in my home assignment. I contains two parts and I don't quite understand what is the difference between them.The question is: Let $n > 2$ be an integer such that ...
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Showing the existence of $O^{\pi}(G)$

Assume G is a finite group. I am trying to show the existence of $O^{\pi}(G)$, the unique normal subgroup of G minimal such that $G/O^{\pi}(G)$ is a $\pi$-group, i.e. a group whose order is divisible ...
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How to find the power of generator defined over finite field , $\mathbb F_{2 ^m}$?

List item Actually ,I am trying to execute the algorithm to find the power of generator of field(group) as shown in table of attached file but when k=7 and onward ,I could not understand what is ...
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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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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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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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group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
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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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103 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 ...