A LOWER BOUND ON THE NUMBER OF FINITE SIMPLE GROUPS

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A LOWER BOUND ON THE NUMBER OF FINITE SIMPLE GROUPS

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Mar 5, 1980 - A LOWER BOUND ON THE NUMBER OF FINITE SIMPLE GROUPS ... Erdos (3) got successively better upper bounds for S(n) by refining an ...

Intrnat. J. Math.

797

Math. Sci.

Vol. 3 No. 4 (1980) 797- 799

A LOWER BOUND ON THE NUMBER OF FINITE SIMPLE GROUPS

MICHAEL E. MAYS Department of Mathematics West Virginia University Morgantown, West Virginia 26506 U.S.A. (Received March 5, 1980)

ABSTRACT.

l{m
>

nl/4/log n.

KEV WORDS AND PHRASES. Spl groW, aymptotic formula. 1980 MATHEMATICAL SUBJECT CLASSIFICATION.

Primary 20 O 06.

The non-.spedialist reader should refer first to Hurley and Rudvalis (4). Write

S(n)

l{m
_ I,

2 or p

and if p

p

a

3 and k

2 then a > i, and

we have

k

f(k,p,a)

q

k(k-l)/2

E

(ql-

l)/(k, p-l)

PSLk(q)

< n.

iffi2

Artln (5) showed that in exactly two cases distinct trlpletons give rise to

isomorphic groups, and in one case there are non-isomorphic groups of the same k(k-l)/2 (k (k+l)/2)-1 < k 2 order in that family. Since f(k,p,a) < q q q

S(n) >> m

l{m

< n:

ak 2 p }I-

S(n) >>

r. affil

there exists (k,p,a) satisfying i), 2), and 3) such that

Such tripletons may be counted by a triple sum, and we have

r.

E i/ak 2 kffi2p_ 2,

NUMBER OF FINITE SIMPLE GROUPS log

s(n)

n/4E

n/aE log 2 12(nllak2),

(log

log 2

799

k-2

a-i

and the Prime Number Theorem using a

1 and k

2 yields S(n) >>

nl/4/log

This theorem is of interest because it has ben conjectured (3) that

o(nl-),

or even S’(n)

o(nl/3).

We have that 1/4

n.

S’ (n)

is a lower bound on the

exponent of n, and if when all simple groups are classified no new family denser

than the projective special linear groups appears, analyzing a perhaps more complicated triple sum carefully should yield the best exponent b in the estimate

S’(n)

o(nb).

ACKNOWLEDGMENT:

I wish to thank Professor Raymond Ayoub of the Pennsylvania

State University for his advice and help. REFERENCES i.

Dornhoff, L. Simple groups are scarce, Proc. Am. Math. Soc. 19(1968) 692-696.

2.

Dornhoff, L. and E. L. Spltznagel, Jr. Density of finite simple group orders, Math Zeltschrlft 106(1968) 175-177.

3.

ErdDs, P. Remarks on some problems in number theory, Math. Balcanlca 4.32 (1974) 197-202.

4.

Hurley, J. and A. Rudvalls. Finite simple groups, Am. Math. Monthly 84(1977)

693-714. 5.

Artln, E. The orders of the linear groups, Comm. Pure and Appl. Math. 8(1952) 355-365.