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Characters of finite quasigroups. III: Quotients and fusion. (English) Zbl 0667.20053

[Part II cf. ibid. 7, 131-137 (1986; Zbl 0599.20110).]
This paper examines two topics which do not extend directly from group- theoretical concepts, since they are respectively trivial or inapplicable in group theory. The first topic concerns the question of how characters of a homomorphic image lift to characters of the preimage. The second topic concerns sets Q carrying two quasigroup structures \((Q,+)\) and (Q,.), such that the multiplication group \(Mlt(Q,+)\) of \((Q,+)\) is a subgroup of the multiplication group \(G=Mlt(Q,.)\) of (Q,.). In part 3 of the paper, the Fusion Theorem shows how the character theory of (Q,.) is determined by the character theory of \((Q,+)\) and the way that (Q,.) “fuses” the conjugacy classes and characters of \((Q,+)\). Many results of this paper generalize readily to the case of a permutation group G and its subgroup H acting on a set Q, in such a way that the centralizer ring V(H,Q) of H on Q is commutative.
Reviewer: C.Pereira da Silva

MSC:

20N05 Loops, quasigroups
20C99 Representation theory of groups

Citations:

Zbl 0599.20110
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Full Text: DOI

References:

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[3] Huppert, B., Endliche gruppen 1, () · Zbl 0412.20002
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[5] Johnson, K.W.; Smith, J.D.H., Characters of finite quasigroups 11: induced characters, Europ. J. combinatorics, 7, 131-137, (1986) · Zbl 0599.20110
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