On the Universality of Words for the Alternating Groups
Copyright policy )On the Universality of Words for the Alternating Groups
We prove the following theorem on the finite alternating groups A n {A_n} : For each pair ( p , q ) (p,q) of nonzero integers there exists an integer N ( p , q ) N(p,q) such that, for each n ⩾ N n \geqslant N , any even permutation a ∈ A n a \in {A_n} can be written in the form a = b p ⋅ c q a = {b^p} \cdot {c^q} for some suitable elements b , c ∈ A n b,c \in {A_n} . A similar result is shown to be true for the finite symmetric groups S n {S_n} provided that p p or q q is odd.
Subgroups of symmetric groups, finite symmetric groups, Generators, relations, and presentations of groups, Symmetric groups, finite alternating groups, even permutation, Simple groups: alternating groups and groups of Lie type
Subgroups of symmetric groups, finite symmetric groups, Generators, relations, and presentations of groups, Symmetric groups, finite alternating groups, even permutation, Simple groups: alternating groups and groups of Lie type
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