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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116 views

trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...
2
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96 views

Schmidt group and maximal subgroups

Let $G$ be a Schmidt group, a minimal non-nilpotent group, so that $G$ is not nilpotent, but every proper subgroup of $G$ is nilpotent. I want to prove $G$ has precisely two classes of maximal ...
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300 views

Characters of double groups

Problem: I want to do some calculations with the character projection operator to investigate the irreducible representations of wave functions. Until now, I did these calculations for simple ...
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68 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
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166 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
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37 views

Separable elements of a finite abelian group

Let $\mathbf G$ be a finite abelian group, let $a, b \in \mathbf G$, and let $\langle a \rangle$ and $\langle b \rangle$ be the cyclic subgroups of $\mathbf G$ generated by $a$ and $b$ respectively. ...
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194 views

Element of a certain order in multiplicative group of residues

Let's say that $G$ is the multiplicative group of residues mod $p$, where $p$ is prime. I know that the order of an element $g \in G$ is the least $k$ for which $g^k \equiv 1 \mod{p}$. How can we go ...
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355 views

$G$ fin ab group, acts faithfully, transitively on $X$, then $|X|=|G|$

Let $G$ be a finite abelian group. Suppose that $G$ acts faithfully and transitively on a set $X$. Show that $|X|=|G|$. Deduce that the action is equivalent to the action of $G$ on itself by left ...
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119 views

Find the cycle index of cyclic group $C_p$ where $p$ is prime

Find the cycle index of cyclic group $C_p$ where $p$ is prime. No idea where to start...any help pointing me in the right direction would be appreciated.
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55 views

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
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52 views

characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a ...
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102 views

Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
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75 views

Group extension analysis

Let $\mathbb{Z}_p\lhd H\leq\text{AGL}(1,p),\mathbb{Z}_q\lhd K\leq\text{AGL}(1,q)$ with $p,q$ prime. Let $G=H.K$. Can one show that $G$ contains a normal subgroup of order $pq$? Note: Here $G=H.K$ ...
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121 views

Lie Theory in Finite Groups

Currently, I have interest in Finite groups. I also want to learn Lie Groups, Lie algebras, and their representations. But I do not have any motivation for it. Question What are simple but ...
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76 views

“unitary group” with respect to non-hermitian matrices?

the group $GU(n,q)$ is usually defined as the group of $n$ by $n$ matrices $X$ over $\mathbb{F}_{q^2}$ such that $X^H X=I_n$, where $I_n$ denotes the identity matrix and $^H$ the hermitian transpose ...
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426 views

How to find the number of orbits

In our lecture notes, i am confused on how to determined the number of orbits with the given permutations in S4. If i consider the permutation (12) . I can also rewrite it as cycle decomposi- tion and ...
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45 views

Invariants of parabolic subgroups

Let $G$ be a finite reflection group acting on $R^n$. Then for each point $x\in R^n$ we can look at its stabilizer $G_x$. Since $G$ is a finite reflection group, its ring of invariants is a polynomial ...
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47 views

Product of two squares in finite groups

Let $G$ be a finite group and $g$ an element of order $n$ in $G$. Assume that $g$ is a product of two squares. Moreover, assume that $n$ and $k$ are coprime. Prove that $g^k$ is also a product of two ...
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46 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

I'm new to algebaric groups, so please accept my apology in advance if I post crazy questions. Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ ...
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76 views

normal subgroup + Socle

I am looking for examples of socle and normal-subgroup relations. If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) ...
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234 views

Normal Sylow subgroups in a group of square free order

If $|G|=n$ with $n$ square free then there exists at least a normal Sylow subgroup? Any suggestion are welcome. Thanks.
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63 views

A bound for the exponent of the Schur multiplier of group G

Let $G$ be a finite group and $M(G)$ be the Schur multiplier of $G$. Show that the exponent $\exp(G)$ of $G$ divides the product of the exponents $\exp(M(S_p))$ of the Sylow $p$-subgroups $S_p$ of ...
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138 views

Automorphisms of a group and cyclic subgroups

I have the following question: Let $G$ be a group. And $A,B\leq G$ two cyclic subgroups with $|A|=|B|=n$. When does an Automorphism $\alpha\in\mathrm{Aut}(G)$ exists such that $\alpha(A)=B$? If ...
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78 views

Connection between: Expand a representation and dual representation

I worked the proof for 3) in this link out, but I have problems with the last step: Let $\sigma$ be a irred. representation of a normal subgroup $H=\langle z\rangle$ of $G$ and $\sigma'$ its dual ...
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78 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
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55 views

Subgroups of semi-direct products arising from fixed-point-free actions

I am interested in subgroups of semidirect products arising from fixed-points-free actions. Suppose you have a group $A$ acting fixed-point-freely on a group $N$. Can we describe completely the ...
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36 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
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131 views

Structure of elementary abelian group by a cyclic group of coprime order

My mind is a little foggy right now, so I would like to ask for some help on this (or an appropriate reference). Suppose $A$ is an elementary abelian $p$-group and $B$ is a cyclic group of order ...
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149 views

Are there 16 or 24 automorphisms of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$?

In this question I said that the automorphism group of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ has 16 elements because If $\varphi$ is one of this automorphism then ...
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82 views

radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or ...
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112 views

a question about elements of permutation groups

Let $n \in \mathbb{N}$ and let $S_n$ denote the permutation group on $n$ letters. I'm trying to figure out what kind of elements $\sigma, \delta \in S_n$ satisfy the following relations: ...
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217 views

Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no ...
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52 views

A question about semidirect product

When we consider the classification of the group G by semidirect product, we need to consider all the homomorphisms from K to Aut(H), Where G=HK and H$\unlhd$G,H$\bigcap$K=1 But by the theorem: ...
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21 views

Calculation of Fixed point set

Let $p,q$ be two distinct primes and let $G_p$ and $G_q$ be nontrivial $p$-group and $q$-group respectively. Let $G = G_p \rtimes G_q$. Embed $G_q \subset U(n_1), G \subset U(n) $ where $U(n_1), ...
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26 views
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Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
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39 views

Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
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22 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
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28 views

If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots \unlhd G_{s} = G $ is a chief series of $ G $.

The $ p $-Fitting Subgroup of $ G $ is the maximal normal $ p $-nilpotent subgroup Of $ G $ and write it $ F_{p}(G) $. If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots ...
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41 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
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44 views

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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36 views

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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34 views

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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25 views

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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36 views

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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90 views

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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60 views

Characterizing the cosets of a cycle of a finite abelian group with a linear combination of floor functions

Prelude Cconsider the finite abelian group $\mathcal G = \prod_{i=1}^A \mathbb Z_{a_i}$ and let $\mathbf s \in \mathcal G$. Let $\mathcal H = \operatorname{grp}({\mathbf s})$ be the subgroup of ...
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19 views

Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
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23 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...
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22 views

Given a space group, how to determine which layer groups are its subgroups?

I am studying various crystals and the two-dimensional materials that could be potentially obtained by cleaving them (isolating a region bounded by two parallel planes). In elucidating the properties ...
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22 views

can say every group that satisfy in maximal permutizer condition then satisfy then permutizer condition

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...