25.85 One Cohomology Group

Let G be a finite group, M a normal p-elementary abelian subgroup of G. Then the group of one coboundaries B^1( G/M, M ) is defined as

B^1( G/M, M ) = { gamma : G/M rightarrow M ; ; ; exists min Mforall gin G : gamma( gM ) = (m^-1)^g cdot m }

and is a Z_p-vector space. The group of cocycles Z^1( G/M, M ) is defined as

Z^1( G/M, M ) = { gamma : G/M rightarrow M ; ; ; forall g_1, g_2in G : gamma(g_1M cdot g_2M ) = gamma(g_1M)^g_2 cdot gamma(g_2M) }

and is also a Z_p-vector space.

Let alpha be the isomorphism of M into a row vector space {cal W} and (g_1, ..., g_l) representatives for a generating set of G/M. Then there exists a monomorphism beta of Z^1( G/M, M ) in the l-fold direct sum of {cal W}, such that beta( gamma ) = ( alpha( gamma( g_1M ) ), ..., alpha( gamma( g_lM ) ) ) for every gamma in Z^1( G/M, M ).

OneCoboundaries (see OneCoboundaries) and OneCocycles (see OneCocycles) compute the group of one coboundaries and one cocyles given a ag group G and a elementary abelian normal subgroup M. If Info1Coh1, Info1Coh2 and Info1Coh3 are set to Print information about the computation is given.

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