165 research outputs found
Skew-Morphisms of Elementary Abelian p-Groups
A skew-morphism of a finite group is a permutation on fixing
the identity element, and for which there exists an integer function on
such that for all . It
has been known that given a skew-morphism of , the product of
with the left regular representation of forms a
permutation group on , called the skew-product group of . In this
paper, the skew-product groups of skew-morphisms of finite elementary abelian
-groups are investigated. Some properties, characterizations and
constructions about that are obtained
Lie theory and coverings of finite groups
We introduce the notion of an `inverse property' (IP) quandle C which we
propose as the right notion of `Lie algebra' in the category of sets. To any IP
quandle we construct an associated group G_C. For a class of IP quandles which
we call `locally skew' and when G_C is finite we show that the noncommutative
de Rham cohomology H^1(G_C) is trivial aside from a single generator \theta
that has no classical analogue. If we start with a group G then any subset
C\subseteq G\setminus {e} which is ad-stable and inversion-stable naturally has
the structure of an IP quandle. If C also generates G then we show that G_C
\twoheadrightarrow G with central kernel, in analogy with the similar result
for the simply-connected covering group of a Lie group. We prove that
G_C\twoheadrightarrow G is an isomorphism for all finite crystallographic
reflection groups W with C the set of reflections, and that C is locally skew
precisely in the simply laced case. This implies that H^1(W)=k when W is simply
laced, proving in particular a previous conjecture for S_n. We obtain similar
results for the dihedral groups D_{6m}. We also consider C=Z P^1\cup Z P^1 as a
locally skew IP-quandle `Lie algebra' of SL_2(Z) and show that G_C\cong B_3,
the braid group on 3 strands. The map B_3\twoheadrightarrow SL_2(Z) which
arises naturally as a covering map in our theory, coincides with the
restriction of the universal covering map \widetilde {SL_2(R)}\to SL_2(R) to
the inverse image of SL_2(Z).Comment: 14 pages, one pdf graphic (minor improvements
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