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- W2081768610 abstract "Let x ϵ S n , the symmetric group on n symbols. Let θ ϵ Aut( S n ) and let the automorphim order of x with respect to θ be defined by γθ(x)= min {k:x xθ xθ 2 ⋯ xθ k−1 =1} where xθ is the image of x under θ. Let α g ϵ Aut( S n ) denote conjugation by the element g ϵ S n . Let b(g; s, k : n) ≡ ∥{x ϵ S n : k γ α g (x) sk }∥ where s and k are positive integers and a b denotes a divides b . Further h ( s , k : n ) ≡ b (1; s , k : n ), where 1 denotes the identity automorphim. If g ϵ S n let c = f ( g , s ) denote the number of symbols in g which are in cycles of length not dividing the integer s , and let g s denote the product of all cycles in g whose lengths do not divide s . Then g s moves c symbols. The main results proved are: (1) recursion: if n ⩾ c + 1 and t = n − c − 1 then b(g; s, 1:n)=∑ i s b(g; s, 1:n−1)( t i−1 (i−1)! (2) reduction: b ( g ; s , 1 : c ) h ( s , 1 : i ) = b ( g ; s , 1 : i + c ); (3) distribution: let D ( θ , n ) ≡ {( k , b ) : k ϵ Z + and b = b ( θ ; 1, k : n ) ≠ 0}; then D ( θ , m ) = D ( φ , m ) ∨ m ⩾ N = N ( θ , φ ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in g s s of any given length is smaller than the smallest prime dividing s iff b ( g s ; s , 1 : c ) = 1. If g = (12 … p m ) t and sk p m then b(g;s,k:p m ) { 0 ±1 ( mod p) ." @default.
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- W2081768610 date "1978-09-01" @default.
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- W2081768610 title "On solutions of “equations in symmetric groups”" @default.
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- W2081768610 doi "https://doi.org/10.1016/0097-3165(78)90076-6" @default.
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