Schur Multipliers of the Known Finite Simple Groups
نویسنده
چکیده
In this note, we announce some results about the Schur multipliers of the known finite simple groups. Proofs will appear elsewhere. We shall conclude with a summary of current knowledge on the subject. Basic properties of multipliers and covering groups of finite groups are discussed in [6]. Notation for groups of Lie type is standard [3], [8]. G' denotes the commutator subgroup of the group G, Z(G) the center of G, Z„ the cyclic group of order n ; other group theoretic notation is standard (see [5] or [6]). MP(G) denotes the p-primary component of the multiplier M(G) of the finite group G. m(G) is the order of M(G) and mp(G) is that of Mp(G). Also, q denotes a power of the prime p. We describe these results in a sequence of theorems. THEOREM 1. m2(G2(4)) = 2, m3(G2(3)) = 3, m2(F4(2)) = 2. In each of these cases, generators and relations for the (unique) covering group are given. THEOREM 2. M(A2(q)) s Z(S(7(3, q)\ i.e. m(SL/(3, q)) = 1. THEOREM 3. Let G be a Steinberg variation defined over a finite field of characteristic p, i.e. G = An(q\ n ^ 2, Dn(q\ n ^ 4, £>4(g), or E6(q). Then Mp(G) = 1 except for M2( A3(2)) s Z2, M3( X3(3)) s Z3 x Z3, M2( ,45(2)) s Z2 x Z2, M2( £6(2)) s Z2 x Z2. THEOREM 4. /ƒ G is a jRee group of type F4, rften m(G) = 1. THEOREM 5. The Tits simple group F4(2)' has trivial multiplier. THEOREM 6. The sporadic groups below have multipliers of the stated orders. AMS 1970 subject classifications. Primary 20C25, 20D05, 20G40; Secondary 20G05, 20F25.
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