The Units of Group-Rings

Higman, Graham

doi:10.1112/plms/s2-46.1.231

Cited by 367 publications

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“…In the present context, every homotopy equivalence is simple since Wh(7r) = 0 for all groups n of order < 4 [7].…”

Section: Surgerymentioning

confidence: 99%

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On 4-manifolds with universal covering space a compact geometric manifold

Hillman

1993

J Aust Math Soc A

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There are 11 closed 4-manifolds which admit geometries of compact type (S 4 , CP 2 or S 2 x S 2 ) and two other closely related bundle spaces (S 2 xS 2 and the total space of the nontrivial RP 2 -bundle over S 2 ). We show that the homotopy type of such a manifold is determined up to an ambiguity of order at most 4 by its quadratic 2-type, and this in turn is (in most cases) determined by the Euler characteristic, fundamental group and Stiefel-Whitney classes. In (at least) seven of the 13 cases, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.1991 Mathematics subject classification (Amer. Math. Soc): 57 N 13.

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“…In the present context, every homotopy equivalence is simple since Wh(7r) = 0 for all groups n of order < 4 [7].…”

Section: Surgerymentioning

confidence: 99%

“…If / is orientation reversing, then trace(/*) = 0, so of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700032006 [7] On 4-manifolds 143…”

Section: * • ( - ; ) •mentioning

confidence: 99%

On 4-manifolds with universal covering space a compact geometric manifold

Hillman

1993

J Aust Math Soc A

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“…Combining the discussion on p. 340 of [6] with Theorem 11 of [1] shows that an iu group matrix for G exists which is not a permutation matrix or the negative of a permutation matrix precisely when G is not any of the groups (i), (ii), (iii). If M is an iu group matrix for G, not a permutation matrix or the negative of a permutation matrix, then MM T is a pdsiu group matrix for G and not the identity since the (i, i) element of MM T is the sum of squares of the integers in row i of M.…”

Section: There Exist Pdsiu Group Matrices For G In Addition To the Idmentioning

confidence: 99%

“…Let x u x 2 , , x n be variables and let X be an n x n matrix whose (i, j) element is x k where k is determined by g k = g^gj 1 . We say X is a group matrix for G. In this paper we study group matrices which have rational integers as elements.…”

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confidence: 99%

Classes of definite group matrices

Thompson¹

1966

Pacific J. Math.

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“…One can then ask the basic question, how much information about the group G can be deduced from the F-algebra R[G]7 A slightly different formulation is to assume that G and //are groups with R[G] isomorphic to R[H] as /^-algebras and to ask what relations exist between G and H. A discussion of the problem with references can be found in Curtis-Reiner [2, p. 262]. There they prove the early result of Higman [4] that if R is the ring of integers and the groups are finite abelian, then isomorphism of the group algebras implies isomorphism of the groups. Later work of Perlis-Walker [8] and Deskins [3] takes up the problem over fields.…”

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confidence: 99%

Commutative Group Algebras

May¹

1969

Transactions of the American Mathematical Society

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Let R be a commutative ring with identity. If G is a group, then one can form the group algebra of G over R which we shall denote by R [G]. One can then ask the basic question, how much information about the group G can be deduced from the F-algebra R [G] [3] takes up the problem over fields. One result is that if we take R to be the rational numbers, then the group algebras determine finite abelian groups.There are difficulties in trying to apply the methods of group representations to other situations, particularly if the characteristic of R is finite. However, if the group algebra is commutative, i.e., the group is abelian, then in case R has characteristic p, one has the Frobenius endomorphism of R [G]. In this paper we show that systematic application of this mapping enables one to deduce fairly broad conclusions about abelian groups with isomorphic group algebras over commutative rings. In fact the group modulo its torsion subgroup is completely determined.To deduce conclusions about a /^-primary component of its torsion subgroup however, we must require that R behaves sufficiently well with respect to the prime p. More precisely, one requires that p is not invertible in R. Under this condition, the maximal divisible subgroup and the Ulm invariants of the /^-primary component are determined. A complete statement of results is contained in the theorem in the last section. In the middle two sections we work over fields of various types and then generalize to arbitrary commutative rings later.

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The Units of Group-Rings

Cited by 367 publications

References 0 publications

On 4-manifolds with universal covering space a compact geometric manifold

On 4-manifolds with universal covering space a compact geometric manifold

Classes of definite group matrices

Commutative Group Algebras

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