360 research outputs found

    Finding involutions with small support

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    We show that the proportion of permutations gg in SnS_n or AnA_n such that gg has even order and gg/2g^{|g|/2} is an involution with support of cardinality at most nε\lceil n^\varepsilon \rceil is at least a constant multiple of ε\varepsilon. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension nn in odd characteristic that have even order and power up to an involution with (1)(-1)-eigenspace of dimension at most nε\lceil n^\varepsilon \rceil for a linear or unitary group, or 2n/2ε2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil for a symplectic or orthogonal group

    Finding involutions with small support

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    We show that the proportion of permutations gg in SnS_n or AnA_n such that gg has even order and gg/2g^{|g|/2} is an involution with support of cardinality at most nε\lceil n^\varepsilon \rceil is at least a constant multiple of ε\varepsilon. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension nn in odd characteristic that have even order and power up to an involution with (1)(-1)-eigenspace of dimension at most nε\lceil n^\varepsilon \rceil for a linear or unitary group, or 2n/2ε2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil for a symplectic or orthogonal group

    Elements in finite classical groups whose powers have large 1-Eigenspaces

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    We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic

    Identifying long cycles in finite alternating and symmetric groups acting on subsets

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    Let HH be a permutation group on a set Λ\Lambda, which is permutationally isomorphic to a finite alternating or symmetric group AnA_n or SnS_n acting on the kk-element subsets of points from {1,,n}\{1,\ldots,n\}, for some arbitrary but fixed kk. Suppose moreover that no isomorphism with this action is known. We show that key elements of HH needed to construct such an isomorphism φ\varphi, such as those whose image under φ\varphi is an nn-cycle or (n1)(n-1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ\Lambda.Comment: 45 page

    The Divisibility Graph of finite groups of Lie Type

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    The Divisibility Graph of a finite group GG has vertex set the set of conjugacy class lengths of non-central elements in GG and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic

    On the frequency of permutations containing a long cycle

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    A general explicit upper bound is obtained for the proportion P(n,m)P(n,m) of elements of order dividing mm, where n1mcnn-1 \le m \le cn for some constant cc, in the finite symmetric group SnS_n. This is used to find lower bounds for the conditional probabilities that an element of SnS_n or AnA_n contains an rr-cycle, given that it satisfies an equation of the form xrs=1x^{rs}=1 where s3s\leq3. For example, the conditional probability that an element xx is an nn-cycle, given that xn=1x^n=1, is always greater than 2/7, and is greater than 1/2 if nn does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups
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