Let P_n be a Sylow p-subgroup of the symmetric group S_n. We investigate the number and sizes of the P_n∖ S_n / P_n double cosets, showing that most double cosets have maximal size when p is odd, or equivalently, that P_n∩ P_n^x=1 for most x∈ S_n when n is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.

Let $χ$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $χ(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of order 69120 such that this proportion is 511/1152.

We give elementary proofs of the following two theorems on automorphisms of a finite group G: (1) An automorphism of G is inner if and only if it extends to an automorphism of every finite group containing G. (2) There exists a finite group, whose outer automorphism group is isomorphic to G. The first theorem was proved by Pettet using a graph-theoretical construction given by Heineken-Liebeck. A Lie-theoretical proof of the second theorem was sketched by Cornulier in a MathOverflow post. Our proofs are purely group-theoretical.

We consider complex characters of a p-group P, which are invariant under a fusion system F on P. Extending a theorem of B & aacute;rcenas-Cantarero to non-saturated fusion systems, we show that the number of indecomposable F -invariant characters of P is greater or equal than the number of F -conjugacy classes of P. We further prove that these two quantities coincide whenever F is realized by a p-solvable group. On the other hand, we observe that this is false for constrained fusion systems in general. Finally, we construct a saturated fusion system with an indecomposable F -invariant character, which is not a summand of the regular character of P. This disproves a recent conjecture of Cantarero-Combariza.

Let G G be a π \pi -separable group with a Hall π \pi -subgroup H H or order n n . For x ∈ H x\in H let λ ( x ) \lambda (x) be the number of Hall π \pi -subgroups of G G containing x x . We show that ∏ d ∣ n ∏ x ∈ H λ ( x d ) n d μ ( d ) = 1 \prod _{d\mid n}\prod _{x\in H}\lambda (x^{d})^{\frac {n}{d}\mu (d)}=1 , where μ \mu is the Möbius function. This generalizes fixed point formulas for coprime actions by Brauer, Wielandt and Navarro–Rizo. We further investigate an additive version of this formula.

The solvable conjugacy class graph of a finite group G, denoted by Γ _sc(G) , is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist x ∈ C and y ∈ D such that ⟨ x, y⟩ is solvable. In this paper, we discuss certain properties of the genus and crosscap of Γ _sc(G) for the groups D_2n , Q_4n , S_n , A_n , and PSL (2,2^d) . In particular, we determine all positive integers n such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of Γ _sc(G) in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of Γ _sc(G) and the commuting probability of certain finite non-solvable group.

Let k(B0) and l(B0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B0 of a finite group G. We prove that, if k(B0)− l(B0) = 1, then l(B0) ≥ p− 1 or else p = 11 and l(B0) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B0) ≥ p− 1 or else p = 11 and G/Op′(G) ∼= C 11 ⋊ SL(2, 5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2 √ p− 1+ 1− kp(G), where kp(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and Sambale. In particular, we fully classify blocks with 16-dimensional basic algebra.

Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

AbstractThe Frobenius–Schur indicators of characters in a real $2$ -block with dihedral defect groups have been determined by Murray [‘Real subpairs and Frobenius–Schur indicators of characters in 2-blocks’, J. Algebra322 (2009), 489–513]. We show that two infinite families described in his work do not exist and we construct examples for the remaining families. We further present some partial results on Frobenius–Schur indicators of characters in other tame blocks.

We call a finite group G ultrasolvable if it has a characteristic subgroup series whose factors are cyclic. It was shown by Durbin–McDonald that the automorphism group of an ultrasolvable group is supersolvable. The converse statement was established by Baartmans–Woeppel under the hypothesis that G has no direct factor isomorphic to the Klein four-group. We extend this result by proving that Aut(G) is supersolvable if and only if G is ultrasolvable or G=H×C_2×C_2 where H is ultrasolvable of odd order. This corrects an erroneous claim by Corsi Tani. Our proof is more elementary than Baartmans–Woeppel's and uses some ideas of Corsi Tani and Laue.

A Sylow p-subgroup P of a finite group G is called redundant if every p-element of G lies in a Sylow subgroup different from P. Generalizing a recent theorem of Maróti–Martínez–Moretó, we show that for every non-cyclic p-group P there exists a solvable group G such that P is redundant in G. Moreover, we answer several open questions raised by Maróti–Martínez–Moretó.

Given a finite abelian group $G$ and cyclic subgroups $A$, $B$, $C$ of $G$ of the same order, we find necessary and sufficient conditions for $A$, $B$, $C$ to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.

Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.

It is a fun game to complete a partial character table of a finite group. We show that one can reconstruct a missing row or column from a given table. The proof relies on deep properties of fully ramified characters. Moreover, we extend a classification of groups with a "large" character degree started by Snyder and continued by Durfee and Jensen.

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace p by a set of primes π and prove a π-version of Dade's conjecture for π-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a π-version of Alperin's weight conjecture previously established by the authors.

Abstract Let 𝑁 be a normal subgroup of a finite group 𝐺. Let N ≤ H ≤ G N\leq H\leq G such that 𝑁 has a complement in 𝐻 and ( | N | , | G : H | ) = 1 (\lvert N\rvert,\lvert G:H\rvert)=1 . If 𝑁 is abelian, a theorem of Gaschütz asserts that 𝑁 has a complement in 𝐺 as well. Brandis has asked whether the commutativity of 𝑁 can be replaced by some weaker property. We prove that 𝑁 has a complement in 𝐺 whenever all Sylow subgroups of 𝑁 are abelian. On the other hand, we construct counterexamples if Z ⁢ ( N ) ∩ N ′ ≠ 1 \mathrm{Z}(N)\cap N^{\prime}\neq 1 . For metabelian groups 𝑁, the condition Z ⁢ ( N ) ∩ N ′ = 1 \mathrm{Z}(N)\cap N^{\prime}=1 implies the existence of complements. Finally, if 𝑁 is perfect and centerless, then Gaschütz’ theorem holds for 𝑁 if and only if Inn ⁢ ( N ) \mathrm{Inn}(N) has a complement in Aut ⁢ ( N ) \mathrm{Aut}(N) .

A finite group G with center Z is of central type if there exists a fully ramified character λ∈Irr(Z) , i. e. the induced character λ ^G is a multiple of an irreducible character. Howlett–Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p 5 . We show that there are no exceptions for p=5 , i. e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann.

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius–Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of g12…gn2=1 with g1,…,gn∈G.

The character table of a finite group G G determines whether | P : P ′ | = p 2 |P:P’|=p^2 and whether | P : Z ( P ) | = p 2 |P:\mathbf {Z}(P)|=p^2 , where P P is a Sylow p p -subgroup of G G . To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.