Maximal subgroups of symmetric groups

Abstract

The cardinality of M shall be denoted by I MI. The support of s e S(M) is defined by spts={xeM:s(x)#x}. The fixed set of s=fs s={xeM:s(x)=x} = M-s spts.IfIMI = X > d,d < Y < X +,defineS(X,Y) ={scES(M): I spts |< Y}. The alternating subgroup shall be denoted A(X). It is well known that the normal subgroups of X(S, Y) are A(X) and the set of subgroups S(X, Z) with d ? Z ? Y [1]. The set of all groups S(X, Y), d ?. Y? X+, together with A(X) is called the set of symmetric groups on the set M. Since S(X, Y) S(X', Y') and A(X) A(X') if and only if X = X' and Y= Y' [1], the notation is unambiguous up to iso-