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On a theorem of Jordan

1954; Mathematical Sciences Publishers; Volume: 4; Issue: 2 Linguagem: Inglês

10.2140/pjm.1954.4.219

1945-5844

Marshall Hall,

Advanced Mathematical Identities

Introduction.In 1872 Jordan [4] showed that a finite quadruply transitive group in which only the identity fixes four letters must be one of the following groups: the symmetric group on four or five letters, the alternating group on six letters, or the Mathieu group on eleven letters.In this paper Jordan's theorem on quadruply transitive groups is generalized in two ways.The number of letters is not assumed to be finite; and instead of assuming that the subgroup fixing four letters consists of the identity alone, we only assume it to be a finite group of odd order.The conclusion is essentially the same as that of Jordan's theorem, the only other group satisfying the hypotheses being the alternating group on seven letters.2. Proof of the main theorem.The theorem is the following: THEOREM 2.1.A group G quadruply transitive on a set of letters?finite or infinite, in which a subgroup H fixing four letters is of finite odd order 9 must be one of the following groups: S 49 S$ 9 A 6 , Ay or the Mathieu group on 11 letters.Case 1. G on not more than seven letters.A quadruply transitive group on 4 or 5 letters must be the symmetric group.On six letters its order must be at least 6 5 4 3, and hence it is A 6 or 5 6 .On seven letters, its index is at most 6 in Sj.As S 7 does not have a subgroup of index 3 or 6, the only possibilities are A 7 and S 7 .In both S 6 and S 7 there are elements of order two fixing at least four letters, and so these groups do not satisfy our hypothesis.To treat the case in which G is on more than seven letters, we begin with two simple lemmas.LEMMA 2.1.Elements a 9 b in a group, satisfying the relations a 2 = 1, b 2 = 1, (ab) s = 1,