### Two Non-Isomorphic Simple Groups Of The Same
Order 20,160

The Annals of Mathematics, 2nd Series, Vol. 1, No. 1/4 (1899-1900), 147-152

This paper was read before the Seminar on Group Theory, held by Professor Moore, March, 1898; and later before the Chicago Mathematical Club held January 21, 1899, at the Univ. of Chicago.

Introduction

It has never been proved that two simple groups of the same order are necessarily holoedrically isomorphic*; nor has any example been heretofore noticed of two simple groups of the same order which are not holoedrically isomorphic, or abstractly identical.

Dickson has proved that all k-ary linear fractional substitution groups in the Galois Field [p^{n}] (p is a prime, n is a positive integer) with determinant unity, are simple; and, among others, he enumerates the ternary linear fractional substitution group of order 8!/2 in the Galois Field [2^{2}].

Since no element of period fifteen and no element of period six could be found in this group, the suspicion arose that this group was not holoedrically isomorphic to the alternating group of degree eight**, a simple group of the same order, containing substitutions of periods fifteen and six.

At Professor Moore's suggestion the investigation is made as to whether or not these two groups are holoedrically isomorphic, and it is found that *no such isomorphism exists*. The present paper furnishes the first direct proof of this theorem, which has recently been corroborated by Mr. Dickson, whose proof will shortly be published.

* Holoedric isomorphism is the only isomorphism that can exist between two simple groups.

** Jordan: Traité des Substitutions, pp. 380-382, and Moore: Math. Ann. Vol. 51, pp. 417-444, have shown that the quaternary linear homogeneous substitution group of order 8!/2 in the Galois Field [2^{1}], and the alternating group of degree eight, both of which are simple, are holoedrically isomorphic.