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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13 views

How does the universal quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
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1answer
42 views

Fulton and Harris exercise 5.7

I would like to know if I'm on the right track: We're trying to derive the character table for $PGL(2,q=p^k)$ from $GL(2,q)$. The table for $GL$ is given in the text. It gives the hint that the ...
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0answers
19 views

Showing the affine transformation is well-defined

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
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22 views

PGL(2,c) is isomorphic to PSL(2,c)

Definition: Projective special linear group $PSL\left ( n,\mathbb{F} \right )=\frac{SL\left ( n,\mathbb{F} \right )} {\left ( Z\left ( GL\left ( n,\mathbb{F} \right ) \right )\cap SL\left ( ...
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0answers
13 views

Language used in projective linear group

In lectures and text on topic of projective linear group, I hear and see the word "factor out" or "quotient out" thrown around a lot. What is the word supposed to mean? If this is vague, I can ...
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1answer
59 views

Rings where $ab=0$ for all elements

Let $R$ be a ring, not necessarily unital, such that $ab=0$ for all $a,b\in R$. Suppose $R$ only has trivial right ideals. Is it true that $R$ has finite order? Are these rings special?
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1answer
29 views

Quotient group with normal subgroup dividing the order of another group [duplicate]

Let G be a group with subgroup H and let $\Omega$ be the set of right cosets of H in G. Show that if G is a group with a subgroup of index n then G has a normal subgroup with index dividing n! ...
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35 views

Irreducible components of tensor product representations.

Let $(\rho,V)$ be an irreducible representation of a finite group $G$, and let $W$ be a vector space. Then clearly $(\rho\otimes\text{Id}_{W},V\otimes W)$ is also a representation of $G$. I would like ...
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1answer
24 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
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0answers
22 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
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0answers
15 views

Inverse of an element in an external direct product

Let $G = \mathbb{Z}_{4}\times S_{5}$ What is the inverse of $\left ( 3,\left ( 1,2 \right )\left ( 3,5 \right ) \right )?$ The inverse of any elements a in $\mathbb{Z}_{4}$ is 4-a. So the inverse of ...
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15 views

Looking for conditions on generator sets of the form $\{a,b \}$ and $\{a,b,c\}$ on the group $\mathbb{Z}_2^3 \rtimes S_3$

Let $G$ be the group $\mathbb{Z}_2^3 \rtimes S_3$ with the natural action of $S_3$ on the coordinates of $\mathbb{Z}_2^3$. I want to know if there are subsets of $G$ of two elements or also 3 elements ...
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1answer
18 views

Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
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1answer
25 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation ...
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1answer
29 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
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1answer
33 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
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1answer
62 views

What is $gnu(18,480)\ $?

Probably, the number of groups of order $18,480$ can only be determined with GAP. But I may be wrong, so the question should not be understood to be purely computational. A better lower bound than my ...
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1answer
25 views

General property of Fitting series

Let $G$ be a finite solvable group. $F(G)$ (Fitting subgroup) is defined to largest normal nilpotent group contained in $G$. Then $F_2(G)$ is defined to be inverse image of $F(G/F(G))$. i.e ...
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2answers
56 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
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37 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
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1answer
40 views

what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

I am looking for smallest example of a group $G$ such that: $G$ is a finite, non-abelian group $G$ is not simple $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $ $H_1, H_2, \dots $ ...
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1answer
28 views

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be ...
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1answer
36 views

number of generators of a semi direct product

Let $G$ be a finite group. Let $g(G)$ be the minimum set of elements of $G$ required to generate the whole group. Suppose that $G= H \rtimes K$ is a semi direct product of two finitely generated ...
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0answers
21 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
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2answers
913 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does "$G$ is abelian" or "$G$ is cyclic" imply "$\text{Aut }G$ is cyclic"?
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1answer
59 views

When is this cyclic representation irreducible?

Let $G$ be a finite group, and let $(\rho, W)$ be a representation of $G$ on $W$. We assume that $W = \bigoplus_i W_i$ is a direct sum of equivalent irreducible representations $W_i$. There are many ...
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1answer
2k views

Existence of normal subgroups for a group of order $36$

Prove that a group of order 36 must have a normal subgroup of order 3 or 9. Let n2 be the number of 2-Sylow subgroups of G (with |G|=36). Then n must be 1 or 3. Let n3 be the number of 3-Sylow ...
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20 views

Proving a conjugation map is an Inner automorphism of a group

Definition: The map $i_{g}:G\rightarrow G$ $h\mapsto g^{-1}hg$ Lemma: $i_{g}$ is an Automorphism of G called an Inner Automorphism. My attempt to prove this is as follows: ...
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1answer
61 views

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$.

Let $G$ be a group of order $120$, let $H≤G$ with $|H|=40$. Prove that there exists $K$ such that $K\unlhd G$, $K≤H$, and $|K|≥20$. I think this is associated with the action of the left coset of ...
2
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1answer
49 views

Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why?

We have the group $G = \langle a, \,b \; | \; a^2b^2ab^{-1}, \, a^3b^4a^{-2}b^{-3}\rangle$. Obviously $G/G' = \mathbb{Z}_2$. Is it true that $G = \mathbb{Z}_2$ or equivalently that $G'=1$? It is ...
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25 views

Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
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1answer
60 views

What is the next term in the jumping champions?

I am interested in the jumping champions of the number of groups of order $n$, where $n$ is cubefree. After calling LoadPackage("cubefree"); GAP gives the ...
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21 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...
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1answer
23 views

The minimal group with Fitting length three has $p$ section in middle?

Let $G$ be a group with Fitting lengt $3$ i.e $$e< F_1< F_2 < F_3=G$$ and $F(G)=F_1$ and $\bar {F_2}=F(G/F_1)$. Assume that for every proper characteristic subgroup $K$ of $G$, Fitting ...
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0answers
11 views

Exponent operation over elements of $G$

I found a definition of an exponent operation over the element of $\mathbb{G}$ in this paper (page 4): $$ (g^a)^{\% b} = g ^{a \text{ mod } b}$$ I couldn't understand the rest of the paper (Decrypt ...
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1answer
59 views

Can we find a non central element of order 2 in a specific 2-group?

Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order ...
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1answer
39 views

Can every group be obtained from a choice of Sylow subgroup for every prime divisor?

The question is almost clear from title: If $G$ is a finite group of order $p_1^{n_1}\cdots p_r^{n_r}$ then is it always possible to choose one Sylow subgroup for every prime divisor of $|G|$ ...
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0answers
39 views

Proof of Cayley's Theorem

This question relates to the link: Cayley's theorem The way I reasoned in showing the map T is a Homomorphism is as follows: Definition: A Homomorphism $\phi: \left ( G,\ast \right ) ...
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2answers
93 views

What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such ...
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2answers
43 views

Automorphism of a group is a group action [closed]

Let G be a group and let $\Omega$ be a set. Then, the $Aut\left ( G \right )$ acts on $\Omega=G$ How can I show that this is true? Thank in advance.
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3answers
41 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 = ...
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0answers
38 views

unfaithful group action [closed]

Let the group $G=GL\left ( n,\mathbb{F} \right )$ and $\Omega$ be the set of all 1-dimensional subspace of $\mathbb{F}^{n}$ Let $\left \langle V \right \rangle \in \Omega$. Define $\left( \langle ...
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1answer
23 views

An algorithm to find a subgroup generated by a subset of a finite group

I'm currently writing a library on python, and now I'm a little bit stuck on how to find a subgroup generated by a subset $S$ of the group $G$. In the case $S = \{a\}\subseteq G$ the problem's easy: ...
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0answers
28 views

About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if ...
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1answer
51 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that ...
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0answers
22 views

references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
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1answer
17 views

Intersection of two subgroups with given information

suppose we know that $G$ is a finite group with order $43200$ and suppose that $H$ is a subgroup of $G$ with order $80$. Furthermore, assume that $K$ is also a subgroup of $G$ such that $[G:K]=1600$. ...
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1answer
24 views

showing a set is a subgroup of a normaliser

Let $H$ be a subgroup of a group $G$ and defined $N_{G}\left ( H \right )=\left \{ g \in G \mid g^{-1}Hg=H \space\ \right \}$ Show that $H$ is a normal subgroup of $N_{G}\left ( H \right ).$ The ...
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1answer
6 views

Specific condition for a map to be isomorphism

Let $G=\left ( \mathbb{R} \space\ \text{where} 0 \notin \mathbb{R},\cdot \right )$ and let r be a positive integer. Define $\phi:G\rightarrow G$ $x \mapsto x^{r}$ Show that $\phi$ is an ...
0
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1answer
15 views

Order of an element in an external direct product

Consider $\mathbb{Z}_{4}\times \mathbb{Z}_{4}=\left \{ 0,1,2,3 \right \}\times \left \{ 0,1,2,3 \right \}$ The element $\left ( 2,0 \right )$ is of order 2 but I cannot figure out why. $2=LCM\left ...