484 research outputs found
Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent
We consider factorizations of a finite group into conjugate subgroups,
for and ,
where is nilpotent or solvable. First we exploit the split -pair
structure of finite simple groups of Lie type to give a unified self-contained
proof that every such group is a product of four or three unipotent Sylow
subgroups. Then we derive an upper bound on the minimal length of a solvable
conjugate factorization of a general finite group. Finally, using conjugate
factorizations of a general finite solvable group by any of its Carter
subgroups, we obtain an upper bound on the minimal length of a nilpotent
conjugate factorization of a general finite group
Structure computation and discrete logarithms in finite abelian p-groups
We present a generic algorithm for computing discrete logarithms in a finite
abelian p-group H, improving the Pohlig-Hellman algorithm and its
generalization to noncyclic groups by Teske. We then give a direct method to
compute a basis for H without using a relation matrix. The problem of computing
a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian
group G is addressed, yielding a Monte Carlo algorithm to compute the structure
of G using O(|G|^0.5) group operations. These results also improve generic
algorithms for extracting pth roots in G.Comment: 23 pages, minor edit
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