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Register a new userFactorizations of group determinant in group algebra for any abelian subgroup
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Authors
Naoya Yamaguchi
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We give a further extension and generalization of Dedekinds theorem over those presented by Yamaguchi. In addition, we give two corollaries on irreducible representations of finite groups and a conjugation of the group algebra of the groups which have an index-two abelian subgroups.
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We give an analog of Frobenius theorem about the factorization of the group determinant on the group algebra of finite abelian groups and we extend it into dihedral groups and generalized quaternion groups. Furthermore, we describe the group determinant of dihedral groups and generalized quaternion groups as a circulant determinant of homogeneous polynomials. This analog on the group algebra is stronger than Frobeniuss theorem and as a corollary, we obtain a simple expression formula for inverse elements in the group algebra. Furthermore, the commutators of irreducible factors of the factorization of the group determinant on the group algebra corresponding to degree one representations have interesting algebraic properties. From this result, we know that degree one representations form natural pairing. At the current stage, the extension of Frobeinus theorem is not represent as a determinant. We expect to find a determinant expression similar to Frobenius theorem.
We generalize the concept of the group determinant and prove a necessary and sufficient novel condition for a subset to be a subgroup. This development is based on the group determinant work by Edward Formanek, David Sibley, and Richard Mansfield, where they show that two groups with the same group determinant are isomorphic. The derived condition leads to a generalization of this result.
Inspired by the Capelli identities for group determinants obtained by T^oru Umeda, we give a basis of the center of the group algebra of any finite group by using Capelli identities for irreducible representations. The Capelli identities for irreducible representations are modifications of the Capelli identity. These identities lead to Capelli elements of the group algebra. These elements construct a basis of the center of the group algebra.
Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $(B,+,cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.
Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
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