Orientably regular maps with Euler characteristic divisible by few primes
TL;DR: It is proved that, apart from a finite number of known exceptions, a non- abelian simple composition factor T of G is a finite group of Lie type with rank n ≤ x.
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Abstract: Let G be a (2, m, n)-group and let x be the number of distinct primes dividing χ, the Euler characteristic of G. We prove, first, that, apart from a finite number of known exceptions, a non- abelian simple composition factor T of G is a finite group of Lie type with rank n ≤ x. This result is proved using new results connecting the prime graph of T to the integer x.
We then study the particular cases x = 1 and x = 2. We give a general structure statement for (2, m, n)-groups which have Euler characteristic a prime power, and we construct an infinite family of these objects. We also give a complete classification of those (2, m, n)-groups which are almost simple and for which the Euler characteristic is a prime power (there are four such).
Finally we announce a result pertaining to those (2, m, n)-groups which are almost simple and for which |χ| is a product of two prime powers. All such groups which are not isomorphic to PSL2 (q) or PGL2 (q) are completely classified.
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(2,m,n)-groups with Euler characteristic equal to -2^as^b
TL;DR: In this article, the absolute value of the Euler characteristic is a product of two prime powers for groups which are not isomorphic to $PSL_2(q) or $PGL_ 2(q)-group.
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$(2,m,n)$-groups with Euler characteristic equal to $-2^as^b$
TL;DR: Those $(2,m,n)$-groups which are almost simple and for which the absolute value of the Euler characteristic is a product of two prime powers are studied.
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Regular maps of order $2$-powers
TL;DR: In this paper, the authors consider the possible types of regular maps of order $2^n, where the order of a regular map is defined as the automorphism group of the map.
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Mark L. Lewis,Gabriel Navarro,D. S. Passman,Thomas R. Wolf +3 more
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Abstract: 1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)
Finite Group Theory
Alexander Lubotzky,Dan Segal +1 more
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TL;DR: In this window, all groups are assumed finite as discussed by the authors, and a number of results of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).
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