Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,494 questions
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Decomposition of the Weil representation of $\mathrm{Sp}(4, p)$ into irreducible representations
What is the decomposition of the Weil representation of $\operatorname{Sp}(4, p)$ into irreducible representations?
The only things I (think I) know is that all the multiplicities involved are 1. ...
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Are the groups of Lie type $\mathfrak{C}_n$ simple for $n=2$?
$\DeclareMathOperator\PSp{PSp}$I'm currently trying to understand the finite simple groups of Lie type, specifically, under what conditions are the groups of Lie type simple.
In particular, I see that ...
- 139
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Subdividing morphisms between objects in $\mathrm{FinGr}$
Given two finite groups $G,H$ with a homomorphism $\phi:G\to H$ the homomorphism can be injective, surjective or neither. This subdivides the morphisms in the category of finite groups $\mathrm{FinGr}$...
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Two non-isomorphic simple groups with the same cardinality whose reduced algebras are isomorphic
Are there two finite simple groups $G$ and $H$ of the same order, non-isomorphic but their corresponding reduced algebra $C_{\mathrm{red}}^*G$ and $C_{\mathrm{red}}^* H$ are isomorphic $C^*$-algebras?...
- 400
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Non-existence of finite simple groups of squarefull order
A positive integer $n$ is called squarefull if all the prime factors exponents of $n$ are at least $2$. Is it possible to prove that there does not exist any finite simple group with squarefull order, ...
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Reference request: Integral motivic cohomology of $BG$
Are there references that compute the integral motivic cohomology $\mathrm{H}^{p,q}(BG,\mathbb Z)$, for $G$ finite cyclic and where $BG$ is defined over the rationals.
I am particularly interested in $...
- 3,366
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Finite groups for which the maximum degree of the prime graph is 2
Does there exist a finite non-solvable and non-almost-simple group satisfying the following conditions?
The degree of every vertex in its prime graph is at most $2$,
If a vertex $p$ in its prime ...
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Linear-algebraic proof of $p$-group fixed-point theorem
Let $P$ be a $p$-group that acts on a finite set $X$. Let $X^P$ denote the $P$-fixed points of $X$. Then:
$$|X|\equiv |X^P|\pmod p$$
There is an easy combinatorial proof of this: $X$ is a disjoint ...
- 2,413
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Search for: natural classes of finite groupoids (and their Cayley graphs) - similar to 15-game
Take a finite group (e.g. $S_n$), take some its elements - generate subgroup - that simple way gives enormous amount of different groups (and actually with generators - hence Cayley graphs).
Question: ...
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Number of subgroups in a semidirect product
Let $G$ be a finite group and $Sub(G)$ denote the number of subgroups of $G$ including the trivial subgroup and $G$ itself. I believe the following is true:
$Sub(H \times K)\leq Sub(H \rtimes K)$ for ...
- 529
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Isomorphism type(s) of minimal groups surjecting on both $G$ and $H$
Fix a prime $p$, and two finite $p$-groups $G$ and $H$. Then we can try to find a group $K$ of minimal order such that there exists a surjective homomorphism $\alpha\colon K\to G$ and another ...
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A character isometry in $A_5$
Background:
Let $G$ be a finite group. Fix a prime $p$. We say that $H\subseteq G$ is strongly $p$-embedded if:
$p$ divides $|H|$;
For every $x\in G-H$, $p$ does not divide $|H\cap H^x|$.
Some facts ...
- 2,413
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Cyclic 2-subgroups of GL(n,Z_p) for n>1
Let $p$ be a prime, $p\neq 2$. Let $Q$ be a cyclic $2$-group and $P$ an elementary abelian $p$-group of rank $n$. Suppose that $Q$ acts faithfully on $P$ (so $Q$ is isomorphic to a subgroup of $GL(n,...
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Classification of finite simple groups of order $4m$, where $m$ is an odd square free integer
Is there any classification of finite simple groups of order $4m$, where $m$ is an odd square free integer?
Some examples are of order $60, 660, 1092, 12180, 102660$.
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Sylow $2$-subgroup of a finite perfect group
Let $G$ be a finite perfect group with non-trivial center. I am trying to show that a Sylow $2$-subgroup of $G$ is non-abelian.
I think one needs to use transfer homomorphisms here. However I can not ...
- 529
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What is the significance of the height of a character?
Let $G$ be a finite group, and fix a prime $p$ which divides $|G|$.
The ordinary (complex) characters $\text{Irr}(G)$ can be partitioned into what are called $p$-blocks, and to each block $B$ can be ...
- 2,413
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Even permutations on geodesic between even permutations $p,q\in \mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\supp{supp}$Let $n\in\mathbb{N}$ and $P(n)$ be the group of even permutations on $n$ symbols and $\SO(n)$ the special orthogonal group. It is well know ...
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Classifying the nature of idempotents on finite groups
Let $A = \ell^1(G)$ be the convolution Banach $*$-algebra of a finite (not necessarily abelian) group $G$ with norm $\|\cdot\|_1$,
let $\mu \in A$ be a probability measure, and set $\delta = \|\mu * \...
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Is there a finite group $G$ with nontrivial Bogomolov multiplier such that $\mathbb{C}^n/G$ admits a crepant resolution?
Let $G$ be a finite group acting linearly on $\mathbb{C}^n$.
A crepant resolution of $\mathbb{C}^n/G$ is roughly one which does not affect the canonical class. See here for a nice discussion of ...
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The generalized Frattini argument
As we all Know, the Frattini argument is that "Let $N$ is a normal subgroup of $G$. If $P$ is a Sylow $p$-subgroup of $N$, then $G=N\cdot N_{G}(P)$". Now, I want to know whether the ...
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Modular representation theory: are these two ideals the same?
Let $k$ be a field of characteristic $p>0$. Let $G$ be a finite group with $p\mid |G|$.
For each conjugacy class $K\in\text{cl}(G)$, let $x_K\in K$ be a representative, let $C_K=\mathbf{C}_G(x_K)$, ...
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Hyperelliptic curve with octahedral symmetry
The hyperelliptic curve (Riemann surface) $y^2=x^8+14x^4+1$ of genus
$3$ has binary octahedral symmetry. The earliest mention of this
curve we found is
Rodríguez, Rubí E.; González-Aguilera, Víctor. ...
- 18.1k
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References for Burnside's theorem without character theory
I'm trying to find a textbook reference for the pure group theory proof of Burnside's $p^a q^b$ theorem, and it's surprisingly difficult to locate one. Surely there must be treatments of the ...
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How does $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ act on $\text{Irr}(G)$?
Let $G$ be a finite group. For a field $F$ (algebraically closed of characteristic $0$), let $\text{Irr}_F(G)$ denote the irreducible characters of $G$ over $F$.
$\text{Gal}(\mathbb{C/R})$ acts on $\...
- 2,413
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On number of subgroups of finite non-abelian simple groups
It is known that there exist non-isomorphic non-abelian finite simple groups with same order. For example one can refer to: Non-isomorphic finite simple groups
My question is: Can there be two non-...
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How can one obtain an inclusion of an induced module and the cokernel thereof with MAGMA?
I would like to ask a MAGMA question.
In the MAGMA code below,
...
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Subgroup structure of $\mathrm{J}_4$
Up to isomorphism, there are two groups which are maximal subgroups of both of the simple groups $\mathrm{M}_{24}$ and $\mathrm{L}_5(2)$ (using ATLAS notation). These have structure $2^4:\mathrm{A}_8$ ...
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A query regarding maximal subgroups of a finite non-solvable group
This is some kind of continuation of an earlier MO post: Existence of maximal subgroups of even order which are not normal
It appeared in a comment in the above post. I believe the following is true:
...
- 529
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RT-structures in finite groups
During my research in Algebraic Geometry, I was led to the following problem in Combinatorial Group Theory, strictly related to finite quotients of pure surface braid groups.
Let $G$ be a finite group....
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Composition factors of induced representations of semi-direct products
As I have not yet recieved any answers, this question is cross-posted from stack exchange
Let $H$ be a subgroup of a finite group $G$ and let $\phi:\mathbb{Z}/2\to \text{Aut}(G)$ such that $\phi(1)(H)=...
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Existence of maximal subgroups of even order which are not normal
Let $G$ be a finite non-solvable group. Does $G$ always have a maximal subgroup of even order which is not normal in G?
Attempt: As $G$ is non-solvable, $|G|$ is even and has an element of order $2$, ...
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Size of Chowla sets
Definition (Chowla subset).
A nonempty subset $S$ of a group $G$ is called a Chowla subset if every element of $S$ has order strictly larger than $|S|$, i.e.
$$\mathrm{ord}(x) > |S| \quad \text{for ...
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Almost unipotent characters
Let us consider a split adjoint simple group $G$ over $\overline{\mathbb{F}_q}$. Then we have a Frobenius map $F$, and we can consider a finite group of Lie type, $G^F$. (We can assume that the ...
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Examples of representations of finite groups with conditions on the tensor square
I would like to have examples of a finite group, $G$, with a finite dimensional representation, $V$, (over the complex numbers, say) with four conditions:
$V$ has a $G$-invariant inner product
The ...
- 570
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A "discriminant" for finite groups
Let $G$ be a finite group, let $H\subseteq G$ be a subgroup, and let $T$ be a set of representatives for the left cosets of $H$ in $G$.
Let $\lambda\in\text{Irr}(H/H')$ be a linear character of $H$.
...
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When is the ring of integers of a character field the ‘character ring’?
Let $G$ be a finite group with an irreducible complex character $\chi$.
Let $\mathbb Q(\chi)$ denote the field extension of the rationals generated by the values of $\chi(g)$ for $g \in G$.
A theorem ...
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When does an $FG$-module have a projective $R$-form?
Let $(F,R,k)$ be a splitting $p$-modular system for a finite group $G$. (Here, $R$ is a discrete valuation ring with residue field $k$ of characteristic $p$ and field of fractions $F$.)
Let $U$ be an $...
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Proof that the unitary associator isomorphism is involutive when the first and last object coincide
Consider the category of finite dimensional unitary representations of some compact group $ G $.
As described here we can define a map $ \Phi_{i,j}^{k,m} $. Now suppose that $ V_i \cong V_k $ (in ...
- 4,577
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Interesting examples of non-isomorphic groups where the probability distribution of the number of square roots of elements is the same
Given a finite group $G$, let $r(g)$ denote the number of square roots of $g$ in $G$, namely:
$$
r(g) = \#\{x\in G \mid x^2 = g\}.
$$
When $g$ is sampled uniformly at random from $G$, $r(g)$ becomes a ...
- 5,894
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Cartan matrices and defect groups of blocks of group algebras
Let $G$ be a finite group, $k$ a characteristic $p$ field, and $B$ a block of the group algebra $kG$ with defect group $D$. If $k$ is sufficiently large (e.g., contains $|G|$-th roots of unity), it is ...
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Sum over all elements in conjugacy class of $S_n$ has all integer eigenvalues in any representation?
Sorry, for the question being obvious or well-known for some, just want to reconfirm not to mislead myself and colleagues. It seems the answer might follow from previous posts by N.Elkies and B....
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Largest 3-zero-sum-free subset in $(\mathbb{Z}/4\mathbb{Z})^n$?
I’m investigating the largest subset $H \subseteq (\mathbb{Z}/4\mathbb{Z})^n$ with no three distinct vectors $x, y, z \in H$ such that $x + y + z \equiv 0 \pmod{4}$ (pointwise addition), as posed by ...
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$\{p\text{-th powers}\} \cap \{q\text{-th powers}\} = \{pq\text{-th powers}\}$ for $p,q$ coprime in a group?
Let $G$ be a finite group, and $p$ and $q$ two coprime positive integers. Let $x \in G$. Assume that $x = y^p = z^q$ for some $y, z \in G$. Is it true that $x = w^{pq}$ for some $w \in G$ ?
Quite ...
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2-periodic resolutions of $C_p$-modules
Let $p$ be a prime and denote by $C_p$ the group with $p$ elements. It is well-known that for any $C_p$-module $M$, the cohomology groups $\textrm{Ext}_{\mathbb ZC_p}^*(\mathbb Z,M)$ are 2-periodic in ...
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Reference request: $p$-local Frobenius complements in finite groups
Let $G$ be a finite group, and let $p$ be a prime.
Let $H\subseteq G$ be a subgroup, where $p$ divides $|H|$. We shall say that $H$ is a $p$-local Frobenius complement if $H\cap H^x$ is a $p'$-group ...
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Do different $R$-forms for the same simple $FG$-module have the same vertex?
Let $G$ be a finite group. Let $R$ be a discrete valuation ring with residue field $k$, where $k$ has positive characteristic $p$. Let $F$ be the field of fractions of $R$.
Let $V$ be a simple $FG$-...
- 2,413
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A lower bound on the number of involutions in nonabelian finite simple groups
As a consequence of the Brauer-Fowler theorem, I am aware of an upper bound on the number of involutions in a nonabelian finite simple group G. Is there any such known lower bound on the number of ...
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Does an irreducible representation $\;p:G\rightarrow (V \rightarrow V)$ always span the whole space of maps $V\rightarrow V$?
EDIT(Andy Putman): Since it's written in what I think is a confusing way, I'm going to rewrite the question in a different language. The original question is below.
Let $G$ be a group and let $V$ be ...
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Strange metrics on finite groups
Let $G$ be a finite group, and fix a prime $p$. For conjugate elements $a,b\in G$, define:
$$d_c(a,b)=\min\{\nu_p(|H|)\mid H\triangleleft G,\text{ there is }h\in H\text{ such that }a^h=b\}$$
where, as ...
- 2,413
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430
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Cup-square obstructions via $j_*: H^k(G, \mathbb{F}_2) \to H^k(G, \mathbb{C}^*) $ for finite groups
Let $G$ be a finite group. Consider the homomorphism $j: \mathbb{F}_2 \to \mathbb{C}^*$ given by $j(1) = -1$. This induces a map in cohomology:
$$
j_*: H^k(G, \mathbb{F}_2) \to H^k(G, \mathbb{C}^*).
$$...
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