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|Title:||Lifting Automorphisms to Stem Extensions of a Finite Group|
|Author(s):||Fry, Michael Donoho|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||A central extension of finite groups e: 0 (--->) A (--->) E (--->) G (--->)1 is said to be a stem extension of G if A is contained in the commutator subgroup E' of E. Schur showed that A must be isomorphic to a subgroup of the finite abelian group M = M(G) = H('2)(G,(//C)('(.))), where ((//C)('(.))) is the group of complex units. In case A (TURNEQ) M, we say that e is a stem cover of G.
An automorphism (sigma) of G lifts to e if there is a commutativediagram e: 0 (--->) A (--->) E (--->) G (--->) 1 (TURNEQ)(DARR) (TURNEQ)(DARR) (DARR)(sigma) e: 0 (--->) A (--->) E (--->) G (--->) 1. The group G is called an group if there is some stem cover of G to which every automorphism of G lifts. One known fact is that if GCD( (VBAR)G/G'(VBAR), (VBAR)M(VBAR)) = 1, then G is . Theorem. (I) If every Sylow subgroup of G is , then G is . (II) If for each prime p dividing GCD( (VBAR)G/G'(VBAR), (VBAR)M(VBAR) , (VBAR)Out G(VBAR) ), some Sylow p-subgroup of Aut G lifts to some stem cover of G, then G is . (Out G is the outer automorphism group of G.) (III) If G is abelian of odd order, then G is . (IV) Suppose G is elementary of order 2('r). Then G is if and only if r (LESSTHEQ) 2.
What sets groups apartfrom non-groups? Two extensions e(,1) and e(,2) of G are isomorphic if there is a diagram e(,1): 0 (--->) A(,1) (--->) E(,1) (--->) G (--->) 1 (TURNEQ)(DARR) (TURNEQ)(DARR) (TURNEQ)(DARR) e(,2): 0 (--->) A(,2) (--->) E(,2) (--->) G (--->) 1.
In case the right-hand map is 1, we say that e(,1) and e(,2)are type 1 isomorphic. The set (GAMMA) of type 1 isomorphism classes of stem covers of G can be made into an Aut G-set in such a way that two stem covers are isomorphic (as extensions) if and only if the corresponding elements of (GAMMA) are in the same Aut G-orbit. Theorem. The finite group G is if and only if the Aut G-set (GAMMA) has a fixed point. If G is , then (GAMMA) can be given an additive structure making it an Aut G-module. In fact, there is a canonical way of making the group K = Ext(G/G',M('*)) into an Aut G-module (M('*) = Hom(M,(//C)('(.)))) and if G is , then (GAMMA) and K are equivalent Aut G-sets.
The property is also characterized by the existence of a "nice"splitting of the Universal Coefficient Sequence 0 (--->) Ext (G/G',M('*)) (--->) H('2)(G,M('*)) (--->) Hom(M('**),M) (--->) 0. Somewhat related to the lifting problem is the notion of isoclinism of extensions. For a given central expansion e: 0 (--->) A (--->) E (--->) G (--->) 1, let U(e) denote the subgroup of Aut G consisting of the autoclinisms of e (isoclinisms of e to itself). If I(,e) denotes the group of automorphisms of G that lift to e, then we obtain a tower I(,e) (LESSTHEQ) U(e) (LESSTHEQ) Aut G.
It is known that everycentral extension of G is isoclinic to some stem extension of G. Theorem. The finite group G is if and only if every central extension e of G is isoclinic to some stem extension e' of G satisfying I(,e)' = U(e') = U(e).
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980.
|Date Available in IDEALS:||2014-12-14|