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All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2G2(q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bit- length O (log n), while SL(n, q) has a presentation with 6 generators, 25 relations and bit-length O(logn + logq). [ABSTRACT FROM AUTHOR]

Let G ≔ G2( K) be a simple algebraic group of type G2 defined over an algebraically closed field K of characteristic p > 0. Let σ denote a standard Frobenius automorphism of G such that Gσ ≅ G2( q) with q ⩾ 4. In this paper we find all reductive subgroups of G and quasi-simple subgroups of Gσ in the defining characteristic. Our results extend the complete reducibility results of [Liebeck and Seitz, Mem. Amer. Math. Soc. 121: 580, 1996, Theorem 1]. [ABSTRACT FROM AUTHOR]

Let G ≔ G2( K) be a simple algebraic group of type G2 defined over an algebraically closed field K of characteristic p > 0. Let σ denote a standard Frobenius automorphism of G such that Gσ ≅ G2( q) with q ⩾ 4. In this paper we find all reductive subgroups of G and quasi-simple subgroups of Gσ in the defining characteristic. Our results extend the complete reducibility results of [Liebeck and Seitz, Mem. Amer. Math. Soc. 121: 580, 1996, Theorem 1]. [ABSTRACT FROM AUTHOR]

If G is a group and H is a subgroup of G we write for the lattice of over-groups of H in G. It is an open question whether or not every finite lattice is isomorphic to some finite group interval lattice , where G is finite. We prove that no DΔ( m1, ... , mt)-lattice and few M-lattices have the form , where G is an alternating or symmetric group of prime degree. [ABSTRACT FROM AUTHOR]

We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of Z ⋊ H. We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than Z ⋊ H. As an application we construct, for each prime power q, a digraph with q – 1 vertices and automorphism group equal to the semilinear group ΓL(1, q), thus proving that ΓL(1, q) is 2-closed in the sense of Wielandt. [ABSTRACT FROM AUTHOR]

We show that a connected split reductive group G over a field of characteristic 0 is determined up to isomorphism by specifying a maximal torus T of G, the set of isomorphism classes of irreducible representations of G, and the character homomorphism from the Grothendieck ring of G to that of T. More precisely, we determine all isomorphisms compatible with the specified data. [ABSTRACT FROM AUTHOR]

Let G be a permutation group acting on a finite set Ω. An uncovering-by-bases (or UBB) for G is a set of bases for G such that any r-subset of Ω is disjoint from at least one base in , where , for d the minimum degree of G. The single-orbit conjecture asserts that for any finite permutation group G, there exists a UBB for G contained in a single orbit of G on its irredundant bases. We prove a case of this conjecture, for when G is k-transitive and has a base of size k + 1. Furthermore, in the more restricted case when G is primitive and has a base of size 2, we show how to construct a UBB of minimum possible size. [ABSTRACT FROM AUTHOR]

Let G be a finite Coxeter group. Using previous results on Weyl groups, and covering the cases of non-crystallographic groups, we show that G has an involution model if and only if all of its irreducible factors are of type A n, B n, D2 n+1, H3, or I2( n). [ABSTRACT FROM AUTHOR]

Let GL and U denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p. Let n be odd, and let χ be an irreducible character of either of these groups which is an extension of a real-valued character of GL or U. Let yτ be an element of GL or U such that ( yτ)2 is regular unipotent in GL or U, respectively. We show that is prime to p and otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way. [ABSTRACT FROM AUTHOR]

Pálfy proved that given a solvable group G and a set of prime divisors of character degrees of G of cardinality at least 3, there exist two different primes such that pq divides some character degree. The solvability hypothesis cannot be removed from Pálfy's theorem, but we show that the same conclusion holds for arbitrary finite groups if . [ABSTRACT FROM AUTHOR]

Let G denote a finite group of permutations of a finite set . Given an orbit function : and a class function : G and their extensions to subsets of and G, consider the quantity m,, which is the maximal value of ()( CG()) over all nontrivial subsets of . Denote by the set of non-empty subsets of which satisfy m()( CG() = m,. Our main result, Theorem 1, states that if contains a unique maximal or minimal element with respect to inclusion, then g = for each g G and CG() is a normal subgroup of G. This result may have potential future applications. As an example of an application of Theorem 1 we prove that if either G is a simple group or it is transitive on , T is a normal subset of G not containing 1, is a G-invariant subset of and , then (see Theorem 6). [ABSTRACT FROM AUTHOR]

In this paper we classify irreducible representations of quasi-simple groups with cyclic Sylow p-subgroup P = 〈 g〉 such that ( g) has at least one eigenvalue of multiplicity 1. [ABSTRACT FROM AUTHOR]

We establish linear bounds for the ρ-σ conjecture for irreducible character degrees and conjugacy class sizes of any finite group. [ABSTRACT FROM AUTHOR]

Various combinatorial equivalents are given to the Lebesgue state in archimedean lattice-ordered groups with order-unit. The proofs use piecewise linear functions on polyhedra with rational vertices. [ABSTRACT FROM AUTHOR]

In Part I it was shown that if G is a p-group of class k, generated by elements of orders , then a necessary condition for the capability of G is that r > 1 and . It was also shown that when G is the k-nilpotent product of the cyclic groups generated by those elements and k = p = 2 or k < p, then the given conditions are also sufficient. We make a correction related to the small class case, and extend the sufficiency result to k = p for an arbitrary prime p. [ABSTRACT FROM AUTHOR]

In 1878, Jordan showed that a finite subgroup of GL( n, ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n; for n ⩾ 71, it is ( n + 1)!, afforded by the symmetric group S n+1. We prove a ‘replacement theorem’ that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n. Our proof relies on known lower bounds for the degrees of faithful representations of each quasisimple group, depending on the classification of finite simple groups, through the use of the bounds for primitive groups that the author has previously obtained. [ABSTRACT FROM AUTHOR]

We discuss finiteness properties of a profinite group G whose probabilistic zeta function PG( s) is rational. In particular we prove that if PG( s) is rational and G has a finite number of non-alternating and non-abelian composition factors in a given composition series, then G/Frat( G) is finite. [ABSTRACT FROM AUTHOR]

The article examines fields, values and character extensions in finite groups. It is stated that the number of irreducible F-valued characters of a finite group G does not need to coincide with the number of conjugacy classes of F-elements of G. It is indicated that in order to get the full Theorem A, it is needed to compare the existence of non-trivial F-valued characters with the existence of non-trivial F-elements of finite groups.

Let G be a finite non-abelian group satisfying Φ( G) = 1 and denote by U the nilpotent residual of G. In this paper, we prove that if G is of odd order then , and if G is of even order not divisible by a Mersenne or a Fermat prime then . These results are best possible and the assumption Φ( G) = 1 cannot be omitted. [ABSTRACT FROM AUTHOR]

In the Kourovka Notebook, Deaconescu asks if |Aut G| ≥ φ(| G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = φ(| G|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/φ(| G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. The problem remains open when G is supersoluble, or nilpotent. [ABSTRACT FROM AUTHOR]

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also establish a necessary condition for the capability of an arbitrary p-group of class k, and some further results. [ABSTRACT FROM AUTHOR]

Presents a study on graphs associated with certain subsets of character degrees. Definitions of the graphs; Properties of the graphs; Results of the graphs.

We study Coxeter groups from which there Is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set T of transpositions. These quotients, denoted here by CY(T), are a special case of the generalized Coxeter groups defined in, and also arise in the computation of certain invariants of surfaces. We use a surprising action of Sn on the kernel of the surjection CY(T)→Sn to show that this kernel embeds in the direct product of n copies of the free group π1(T), except when T is the full set of transpositions in S4. As a result, we show that each group CY(T) either is virtually Abelian or contains a non-Abelian free subgroup. [ABSTRACT FROM AUTHOR]