A note on involution centralizers in black box groups. (English) Zbl 1466.20011
A black-box group is a group \(G\) whose elements are encoded by bit strings of length \(\beta\), and group operations are performed by an oracle (the “black box”); the only group operations admissible are multiplication, inversion, and checking for equality with the identity element. Permutation groups and matrix groups defined over finite fields are covered by this model. The upper bound on the order of \(G\) given by \(|G| \leq 2^{\beta}\) shows that \(G\) is finite.
In [Arch. Math. 74, No. 4, 241–245 (2000; Zbl 0956.20022)], J. L. Bray provided a method for calculating centralizers of involutions in black-box groups with an order oracle. In the article under review, the authors refine Bray’s method and provide some interesting applications of their results. In particular, they investigate the action of the Lyons’ sporadic group via conjugation on the set of its involutions. They also perform some calculations in \(E_{6}(2)\), \(E_{8}(2)\) and in \(\mathrm{GL}_{670}(2)\).
In [Arch. Math. 74, No. 4, 241–245 (2000; Zbl 0956.20022)], J. L. Bray provided a method for calculating centralizers of involutions in black-box groups with an order oracle. In the article under review, the authors refine Bray’s method and provide some interesting applications of their results. In particular, they investigate the action of the Lyons’ sporadic group via conjugation on the set of its involutions. They also perform some calculations in \(E_{6}(2)\), \(E_{8}(2)\) and in \(\mathrm{GL}_{670}(2)\).
Reviewer: Egle Bettio (Venezia)
MSC:
| 20D08 | Simple groups: sporadic groups |
| 20E45 | Conjugacy classes for groups |
| 20-04 | Software, source code, etc. for problems pertaining to group theory |
Keywords:
black-box groups; centralizer of involutions; sporadic group; computations in finite groupsCitations:
Zbl 0956.20022Software:
ATLAS Group RepresentationsReferences:
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