Questions tagged [finite-groups]
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.
11,548
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Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
- 1,453
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votes
2
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419
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A modification of a cyclic group that seems to break being a group. What is it?
Background: I came up with this trying to answer an actually silly question of "when can
$1+1=3$ be true" ?
Consider a set $\mathcal S = \{0,\cdots, N-1\}$
coupled with an operation "+&...
- 25.4k
2
votes
2
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112
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Why is a Sylow 5-subgroup abelian?
For weeks I tried to solve the following question on Brilliant:
Fill in the blank: "Every group of order ___ is abelian."
And these are the possible answers I get: 15, 16, 20, 21, 27.
Using ...
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"Simple" group of order $1004913$ problem, fixed point part
Let $G$ be a group of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$. We suppose that $G$ is simple. We want to obtain a contradiction.
This is the Exercise 29 in Chapter 6.2 of Dummit-Foote. As ...
- 405
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48
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About Frobenius Groups of order 1029
In the list of groups in GAP of order $1029=7^3\cdot 3$, there are two, with structure description $U_3(\mathbb{F}_7)\rtimes C_3$.
(Among $19$ groups $G[1], G[2], \ldots, G[19]$ of order $1029$, the ...
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27
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Constructing central extensions or Schur cover of U4(3) in GAP
Part of my group theory project involves we're looking at the group $U_4(3)$, which has abnormally large Schur Multiplier (36) and large automorphism group ($D_{12}$). I need to work with the central ...
- 105
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39
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Character tables of isomorphic groups
Let us consider two groups $G$ and $G'$ which are isomorphic to each other. Since isomorphic groups can be considered as same upto isomorphism, is the character table same for both groups?
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Find an element of order $p+1$ in $GL_2(\mathbb{Z}/p\mathbb{Z})$ [duplicate]
The question says it all. I'm looking for an element of order $p+1$ in $GL_2(\mathbb{Z}/p\mathbb{Z})$. I have already computed the order of the group (so I know it's possible that one exists). I have ...
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How to show that a subgroup of a product of finite cyclic group has itself this property?
Let $G$ be a subgroup of a finite product of finite cyclic groups. Is it easy to prove that $G$ is itself a finite product of cyclic groups without appealing to the structure theorem of finite abelian ...
- 1,476
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A combinatorial property for finite cyclic groups
I was reading about the Cauchy–Davenport inequality and other results in combinatorics and the following property came to my mind.
Let $G$ be a finite group. I call here $S=\{s_1,\ldots,s_n\}\subseteq ...
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Does $S_n$ always embed into $GL_{n-1} (\mathbb{F}_p$)?
$S_n$ is the symmetry group of the standard $n-1$-simplex, which is the convex hull of the standard basis vectors in $\mathbb{R}^n$. One can orthogonally project this shape onto the plane $x_1 +...+ ...
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Modules over the ring $\mathbb{F}_p[C_p]\cong \mathbb{F}_p[X]/(x^p-1)$
I would like to understand the category of modules over the group algebra $\mathbb{F}_p[C_p]\cong\mathbb{F}_p[X]/(x^p-1)$.
I am interested in computing the group cohomology of $C_p$ with coefficient ...
- 1,805
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In a group of order 120 with a normal subgroup of order 5, every subgroup of order 15 contains that normal subgroup. Simplest proof?
Let $G$ be a group of order $120$ with a normal subgroup $N$ of order 5. Let $H$ be any subgroup of $G$ of order $15$. Prove $N$ is a subgroup of $H$.
I have a proof (see below), but I am wondering: ...
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+50
Automorphisms for direct products of finite commutative nilpotent rings.
Let $(R, +, \cdot)$ be an associative commutative nilpotent ring of cardinality $2^n$ such that
$$
r^2 = 0,
$$
for every $r\in R$. Also $(V, +)$ is a vector space over $\mathbb{F}_2$. Let $\...
- 1,344
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A question about coprime action on a 2-group
I'm stuck somewhere in the following problem in Isaacs Finite Group Theory [4D.4], I would appreciate if you could help:
Problem: Let $A$ act via automorphisms on $G$ , where $G$ is a $2$-group and $A$...
- 407
2
votes
2
answers
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Does it mean $|N_H|=|H|$?
Let $G$ be a finite group and $H$ be a proper subgroup of $G$. Let $N_H:=\{g\in G: gHg^{-1}=H\}$ be the normalizer of $H$. I try to show that the index of $H$ in $G$ is greater than the index of $N_H$ ...
- 1,271
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Must a subgroup of $H\times K$ be the direct product of two subgroups of $H$ and $K$ respectively [duplicate]
I have some doubt in the following statement: Let $H$ and $K$ be two finite Groups and let $G$ denote their direct product. If $S＜G$ , then does $S$ must be of the form $H'\times K'$ for some ...
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A Question About Coprime Actions
I'm dealing with the following problem in Isaacs Finite Group Theory [4D.3], I would appreciate if you could help:
Problem: Let $A$ act via automorphisms on $G$ , where $(\vert G \vert, \vert A \vert )...
- 407
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Doubt regarding the generality of an equivalent definition for Abelian Group [duplicate]
I was given a problem as follows:
Let $G$ be a finite group of odd order. If $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b\in G$ then $G$ is abelian.
I was able to show that this is ...
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Problem 1D.2 from Martin Isaacs's Finite Group Theory
I try to solve the following problem:
Fix a prime $p$, and suppose that a subgroup $H \subset G$ (of a
finite group $G$) has the property that $C_G(x) \subset H$ for every
element $x \in H$ having ...
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39
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Find a subgroup $K$ to complete the pullback diagram $G/g_1Hg_1^{-1}\leftarrow G/H\to G/g_2Hg_2^{-1}$.
EDIT: I have realised I made a mistake when decompsoing the morphisms of $\mathscr B_G$.
Nevertheless, the question seems to be interesting on its own, so i will leave it. I would also like to cite ...
- 1,805
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2
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47
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Frattini subgroup of a quotient
Let $G$ be a finite group. The Frattini subgroup $\Phi(G)$ is the intersection all proper maximal subgroups.
If $K \lhd G$ is a normal subgroup, then it is easy to see that $\Phi(G) K/K \leq \Phi(G/K)$...
- 5,193
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Show $\left(F\left[X_{1}, \ldots, X_{n}\right]\right)^{A_{n}}$ is an integral extension of $\left(F\left[X_{1}, \ldots, X_{n}\right]\right)^{S_{n}}$
Let $F$ be a field of characteristic not 2 and let the symmetric group $S_{n}$ act on the polynomial ring $F\left[X_{1}, \ldots, X_{n}\right]$ by permuting the variables, for $n \geq 2$. Let $A=\left(...
- 1,123
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Explicit 3-cocycles for D4
I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{4},C^×)$, where
$$D_4=⟨r,s∣r^4=s^2=e, rs=sr^3⟩$$
is the dihedral group of order 8. I know that there is a ...
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3
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Uniqueness of subgroups of cyclic groups and number of solutions of an equation
In an exercise I am asked to prove the following:
Show that if $G$ is a finite cyclic group with order $n$, the equation $x^m=e$ has $m$ solutions for all $m\mid n$, where $e$ is the neutral element.
...
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If $X$ is a finite $G$-set and $|g|=p^n$ with $n\ge 1$, show that $|X|=|X^{g^{p^{n-1}}}|\pmod{p^n}$.
Let $G$ be a group and let $X$ be a finite $G$-set.
Assume that $g\in G$ is of finite order $p^n$, where $p$ is a prime number and $n\ge 1$.
Let $h=g^{p^{n-1}}$.
Show that $|X|=|X^h|\pmod {p^n}$,
...
- 2,387
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In a finite reflection group, an involution is a product of commuting reflections
I am working through the book Reflection Groups and Coxeter Groups by Humphreys. I got stuck while trying Exercise 1.12.3:
If $w \in W$ is an involution, prove that $w$ can be written as a product of ...
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Is this a valid group structure (of order 12)?
Let
$$
G = \big\langle \, a, b, c \colon a^3 = b^2 = c^2 = (ab)^2 = (bc)^2 = 1, ac = ca \big\rangle. \tag{0}
$$
That is, $G$ is a group that has elements $a$, $b$, and $c$ (though these are not the ...
- 25.8k
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Suppose that $|x| = n$. Find a necessary and sufficient condition on $r$ and $s$ such that $\langle x^r\rangle\subseteq\langle x^s\rangle $
I am doing this exercise from Gallian's Contemporary Abstract Algebra 8th edition and came across this problem. The solution I found in a manual was that $|x^r|$ must divide $|x^s|$. My solution was ...
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To prove that U(20) is not cyclic. [duplicate]
A similar question already has an answer elsewhere on this site, though this question is a bit different from those.
When I first saw the formulation of question, namely:
Show that $U(20)\neq \langle ...
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Lyndon-Hochschild-Serre for the product
General case I'm interested in:
let $G = H_1 \times H_2$ where both $H_i$ are abelian groups. Let $M \in \operatorname{Mod}_{k[G]}$, then one has the spectral sequence $H^i(H_1, H^j(H_2, M))$ ...
- 1,186
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votes
1
answer
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views
General linear group inclusion
Do we have $\operatorname{GL}(n,F)\le \operatorname{O}(2n,F)$ where O means general orthogonal group and $F$ is an algebraically closed field?
I checked some finite group cases: $\operatorname{GL}(2,5)...
- 1,453
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Confusion regarding matrix representation properties
I am studying about matrix representation of finite groups. If the group is defined as
\begin{equation}
G=\{e,a,b,c,.....\}
\end{equation}
then the matrix representation is defined by the collection ...
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Simplicity of high-order groups [duplicate]
Let $G$ be a group of order
$$|G|=\prod^n_{i=1}p_i$$
where each $p_i$ is a prime number and
$$n\geq2\qquad \text{and} \qquad k\neq j\implies p_k\neq p_j \qquad \text{and} \qquad p_1<p_2<\cdots&...
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Manual Procedure for Generating Group
I was trying to come up with groups and came up with the following, but looking at Group Explorer I do not see this group. I came up with this group by building out its Cayley diagram and then writing ...
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Uniqueness of characters of transitive actions
Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then, $|X| \big| |G|$ by the orbit-stabilizer theorem. Let this action induce a permutation representation $\rho: G \to ...
- 1,953
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1
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2nd order Commutant of permutation matrices
Let $d\ge 2$ denote an integer, let $S_d$ denote the group of permutations of $d$ elements, and let $L(\mathbf{C}^d)$ denote the space of linear operators on the Hilbert space of $d$-dimensional ...
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Finite subgroups of O(3)
My question is the following: What are the finite subgroups of O(3), the group of linear isometries?
I managed to find a lot of good references describing the finite subgoups of SO(3) (only direct ...
- 1,544
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How many groups with $100$ elements exist, where for min. $80\%$ of the pairs $(a, b)$ is $ab = cba$, for some constant $c$? [closed]
Got the idea from skew symmetry of matrice rings, where when you swap 2 elements you need to add a minus sign. But I was wondering to what extend is this possible for only a finite group. I guess I ...
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A different approach to proving a property of nilpotent injectors in solvable groups
Let $G$ be a finite solvable group. Call $J\subseteq G$ a nilpotent injector if it is a nilpotent subgroup that contains $\mathbf{F}(G)$, and that is maximal with this property (not properly contained ...
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41
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Theorem Reference Query: $p'$-Elements in Supersolvable Groups
I have the following question: In the context of a finite supersolvable group $G$, where $p$ represents the largest prime divisor of $|G|$, I would like to inquire about a specific theorem, for which ...
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33
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Cartesian product of transitive group actions, stabilizer groups
Consider a finite group $G$ and subgroups $\{H_i\}_{i=0,\ldots,n-1}$ (defined up to conjugacy) such that $G$ acts transitively on each $G/H_i$ [edit: in the canonical way]. Consider the induced action ...
- 363
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answers
85
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Motivation for triple cover of $A_6$ (and $A_7$)
In finite simple groups by Wilson, he constructs the triple cover of $A_6$ by considering the action of the subgroup of $A_6$ preserving the partition $\{12, 34, 56\}$ on the two vectors $(0, 0, 1, 1, ...
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1
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23
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Finding Matrix Representations of a Dicyclic Group
Consider the dicyclic group with 16 elements,
$$
Dic_4 = \langle a,b \mid a^8 = 1, b^2 = a^4, b^{-1} a b = a^{-1} \rangle.
$$
There are two faithful irreps of this group, say $\delta_1$ and $\delta_2$....
- 551
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1
answer
65
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Question on the classification of groups of order 102
I have a question regarding the classification of groups of "small" order; we'll take groups of order $102=2 \cdot 3 \cdot 17$ as an example.
Let G be a group of order 102, note that $P_{17}\...
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votes
1
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81
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How many non-isomorphic groups of order $5832 = 2^3 \cdot 3^6$ are there?
I'm afraid I can't provide much motivation other than personal interest. I have found David Burrell's very recent Ph.D. thesis, which identified a transcription error that resulted in an incorrect ...
- 24.8k
3
votes
1
answer
29
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Group algebra is a free module of finite rank over any group sub-algebra
Let $G$ be a finite group and let $H$ be a subgroup. Let $\mathbb CG$ and $\mathbb CH$ be the corresponding group algebras. Clearly, the injective group homomorphism $H\to G$
defines a $\mathbb CH$-...
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1
answer
125
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Finite group generated by two finite subgroups
Question
Let $G=\langle H,K\rangle$ where $H,K$ are two finite subgroups of $G$. If $H$ is subnormal in $G$, then show that $G$ is finite.
Attempt
I know that $D_{\infty}=\mathbb{Z}_2\ast\mathbb{Z}_2$ ...
- 4,913
0
votes
1
answer
47
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Finding the subgroups of a finite group [duplicate]
I have a question on my math homework that asks me to find 3 subgroups of Z(mod10) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
I understand that the subgroups must hold under the associative, identity, inverse, ...
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76
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Is there a way to computationally verify that the sporadic groups are simple?
I'm trying to understand the "easy" direction of the CFSG: namely, the proofs that the 18 infinite families and 26/27 sporadic groups are indeed simple. I'm working through Simple Groups of ...
- 51