# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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### Smallest $\mathbb R$-algebra which contains a subgroup isomorphic to $A_4$

$A_4$ (the alternating group on $4$ elements) can be thought of as the group of direct Euclidean isometries of a regular tetrahedron. This shows that there is a subgroup of the algebra of $3\times3$ ...
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### Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G. More generally here (MO275769) Qiaochu Yuan ...
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### On the density of the orders excluded by the Sylow theorems for simple groups

If $G$ is a finite group whose order is divisible by a prime $p$ and $p^r$ is the maximal power of $p$ that divides it, the Sylow theorems tell us that the number $n_p$ of Sylow $p$-subgroups of $G$ ...
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I would greatly appreciate it if you kindly give me some advice to tackle the below situation. Let $G$ be the non-abelian 2-group $$G=\langle a, b, c \mid a^{2^n}=b^{2^m}=c^2=1, [a,b]=c, [a,c]=1, [b,c]... 1answer 111 views ### How do I find hyperbolic generating triples for a group using GAP? Let G be a finite group and x, y, z \in G. A hyperbolic generating triple for G is a triple (x, y, z) \in G\times G\times G such that \frac{1}{o(x)}+\frac{1}{o(y)}+\frac{1}{o(z)} <1, \... 4answers 966 views ### \operatorname{PSL}(2,\mathbb{F}_p)  does not embed in \mathfrak{S}_p for p>11 A famous result of Galois, in his letter to Auguste Chevalier, is that for p prime >11 the group \operatorname{PSL}(2,\mathbb{F}_p)  does not embed in the symmetric group \mathfrak{S}_p. ... 2answers 571 views ### The finite groups with a zero entry in each column of its character table (except the first one) \DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}Consider the class of finite groups G having a zero entry in each column of its character table (except the first one), i.e. for all g \... 0answers 99 views ### Tensor products of irreducible representations of GL_{2}(\mathbb{F}_{q}) Throughout the post G = GL_{2}(\mathbb{F}_{q}) where q is a prime power with the prime not being 2. Let V_{1} and V_{2} be cuspidal representations of G over \mathbb{C}. I can understand ... 0answers 137 views ### Subalgebra of group algebra generated by idempotents Let G be a finite group, and let A and B be two abelian subgroups of G. Let K be a number field such that all characters of A and of B take values in K. Let \mathcal{O}_K be the ring ... 1answer 237 views ### How small can the support of a nontrivial \mathbb F_p-cocycle on C_p be? Let p be a prime, and let \phi : C_p^n \to \mathbb F_p be an \mathbb F_p-valued n-cocycle on C_p (the cyclic group of order p) which is not an n-coboundary, i.e. \phi represents a ... 0answers 60 views ### The number of orbits of a two-point stabilizer of the symplectic group Sp(2m,2) I am trying to figure out the number of orbits of a two-point stabilizer of the action of Sp(2m,2) on its two orbits \Omega^+ and \Omega^- as detailed in Dixon and Mortimer's "Permutation ... 2answers 333 views ### Why are finite simple groups useful? [duplicate] The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of ... 3answers 427 views ### Small simplicial set models for BG Let F be a finite group. Is there a model for BF as a simplicial set such that the number of nondegenerate n-simplices grows at most polynomially? For example the Bar construction has the ... 1answer 150 views ### Irreducible representations of finite p-groups Let G be a finite p-group. What are irreducible representations of G over a field of characteristic q, such that (p,q)=1 ? Can we say something in general ? In particular, if there exists ... 0answers 107 views ### Another question concerning finite metacyclic groups Given a non-split finite metacyclic group H, does there always exist a finite split metacyclic group G with a normal cyclic subgroup N of prime power order such that H \cong G/N? Based on my ... 1answer 153 views ### Structures of subgroups of a finite abelian p-group \newcommand\la{\langle}\newcommand\ra{\rangle}Let G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r} with i_1\leq\ldots\leq i_r be a finite abelian p-group. Then there can be many ... 1answer 203 views ### Finite simple groups with the same numbers of elements of orders p and q Let G be a nonabelian finite simple group, and let p and q be distinct prime divisors of the order of G. Is it true that the number of elements of G of order p never equals the number of ... 1answer 181 views ### A transitive action on a specific set Let G be a finite group and \lambda\in G, consider a set$$D^{p+*}_{G}(\lambda):=\{P\in S^{p+*}_G|P^{\lambda}=P=[P,\lambda]\}$$where: S^{p+*}_{G} denotes the set consisting of all non-trivial ... 3answers 443 views ### Large product-1-free sets in finite groups \DeclareMathOperator\SmallGroup{SmallGroup}Definition. A subset A of a group G is called product-1-free if for any sequence of pairwise distinct elements a_1,\dots,a_n of A the product a_1\... 1answer 113 views ### Subgroups and representations of finite groups of Lie type Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition? ... 0answers 88 views ### Action of diagonal automorphisms on the set of irreducible characters of D_n(q) Let S be D_n(q) where q is a prime power. We know that a diagonal automorphism \phi_h of S is of the form g\mapsto hgh^{-1}, where h\in \hat H and \hat H normalizes S. Note that \... 0answers 115 views ### Finite simple groups of automorphisms of finite simple Lie algebras I begin by briefly recalling some basic facts in order to pose my question in context. According to the classification, the finite simple groups are cyclic of prime order, are alternating on n \geq 5... 1answer 321 views ### Number of 1's in binary expansion of a_n = \frac{2^{\varphi(3^n)}-1}{3^n} My question is about the Hamming Weight (or number of 1's in binary expansion) of a_n = \frac{2^{\varphi(3^n)}-1}{3^n} A152007 For example, a_3 = 9709 = (10110111101001)_2  has nine 1's in binary ... 3answers 462 views ### Product-one sets in non-commutative groups A nonempty subset D of a group G is called \bullet decomposable if D\subseteq DD, that is every element x\in D is can be written as the product x=yz of some elements y,z\in D; \bullet ... 1answer 205 views ### Finite groups with a dihedral maximal subgroup Suppose G is a finite group with a dihedral maximal subgroup. Suppose that G is not isomorphic to \operatorname{PSL}(2,q) for some any prime-power q. Is G always solvable? 1answer 151 views ### Is there always a simple module whose Green correspondent is a simple module under some conditions? Let G be a finite group and KG its group algebra over some field K with \mathrm{char}\ K dividing the order of G. It's well-known that the Green correspondence is compatible with the Brauer ... 1answer 255 views ### Is the fixed subring a symmetric algebra? Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A,... 0answers 68 views ### Is Broué's abelian defect conjecture true for finite groups with abelian TI Sylow p-subgroups? I am now interested in Broué's abelian defect conjecture and I have read many papers concerning it. For a prime p, I informally define a finite group to be a p-ATI-group if it has abelian Sylow p... 1answer 288 views ### When is the augmentation ideal projective as RG-module? Let G be a finite group and let R be a commutative ring. I'd like to ask, if there is a theorem of the following kind: The augmentation ideal I_G is projective as RG-module, if and only if ... ?... 0answers 177 views ### Inverse Galois problem on simple groups Im trying to find a way to connect a possible solution of the inverse Galois problem on simple groups to a more general solution on any finite group. I've tryied to mess with the embedding problem for ... 1answer 259 views ### A question about the possibilities of GAP Let R=\mathbb{Z}/1024\mathbb{Z} and G=GL(3,R). Let H be the subgroup of G consisting of all matrices with determinant 1 which are congruent to the identity matrix modulo the ideal 4R. Let ... 0answers 454 views ### Can we generalize the concept of "characters" in group theory via methods from statistics and probability theory? \DeclareMathOperator\Cov{Cov}Motivation: If G is a finite group and \phi=X+iY: G\to \mathbb{T} is a character of G, then \Cov(X,Y)=0 where X, Y are considered as two real random ... 3answers 278 views ### Generation of permutation groups by fixed elements subgroups Suppose (H,X) is a permutation group (with H a group acting faithfully on the set X). Under what circumstances is H generated by its subgroups H_x, where H_x is the subgroup of H fixing ... 0answers 79 views ### Group where Out(G) acts differently on conjugacy classes and irreps? [duplicate] \def\Conj{\mathrm{Conj}}\def\Irrep{\mathrm{Irrep}}\def\Out{\mathrm{Out}}Let G be a finite group, let \Conj(G) be the set of conjugacy classes of G, let \Irrep(G) be the set of isomorphism ... 6answers 787 views ### Are there always more conjugacy classes in the kernel of a morphism to Z_2 than not? Let G be a finite group and let \phi:G\to Z_2 be a homomorphism to the group with two elements. Is it always the case that there are more conjugacy classes in the kernel of \phi than conjugacy ... 0answers 77 views ### Are these maps, associated to finite simple graphs, interesting? Given a finite simple graph on n vertices, say G = (V,\, E), where$$ V = \{ v_1, \ldots , \, v_n \} $$and$$ E \subseteq \{ (v_a, \, v_b) \, | \, 1 \leq a < b \leq n \},$$does there exist a ... 1answer 138 views ### Local vs global nilpotence class (Lazard correspondence) The Lazard Correspondence is often phrased (for simplicity) for p-groups of nilpotence class c < p, but it works more generally whenever every 3-generated subgroup has nilpotence class < p,... 0answers 121 views ### Intersection of \mathrm{PGL}_2(q_0)'s in \mathrm{PGL}_2(q_0^3) \DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}I would like an explanation for some strange numerology which I encountered when studying intersections of subfield subgroups in \PGL_2(q). ... 1answer 110 views ### The \{2,3\}-groups with a condition about \mathbb{C}-characters Let G be a \{2,3\}-group and \lvert G\rvert=2^\alpha\cdot3^\beta. For p\in\{2,3\}, define$$ \nu_p(G)\mathrel{:=}\min\left\{\log_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)_p \mathrel{\...
In all that follows, let $k$ be a field and $G$ be a finite group. It is well-known that the order of $G$ is invertible in $k$ iff the group ring $k[G]$ is semisimple, which is equivalent, inter alia, ...