By Steinke G. F.

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**Example text**

Is a completely I n, p7 i = = max{dl(G/D I i = j) 1, ... ,rn}. 6. 2, then G / Di ~,33 and cll( G / D i ) ~ 2. ·If each . k· . p7 i is 2, then dI(G) ~ 4,~ (3(2) ~ (J(e) ~ (J(n). Thus some]1/ is at least 3. nse G acts symplectically on '. BOUNDS FOIl LINEAll GROUPS EdT. Thus if each , Consequently, some p7 i p/k' Sec. 3( n). ' is at least 4. '\ when n = 8, see p. 156) that C IIf( G) must have trivial center, whence the G IF( G)-module F( G)I Z cannot be irreducible and primitive. Indeed, the hound given in'[Di 1] fora linea~ group of degree 8 is 7 ~ We claim that, for some j, z, k i = 1 or p7 = 22.

It again follows that Proof. 10. (i) Let Va be an irreducible E-submodule of V. Since V is quasi-primitive, VE ~ Va EB· .. 9, Ca(E) is a cyclic normal subgroup of G and thus CG(E) = Cp(E) , . C[G) niodules Vi (see Pro- = T. Since F. = ET, it follows that T = Z(F) = Co(E) is cyclic. 4). Likewise, (ii) Observe that IDjZ(D)1 = IEjZI = e2 . Hence the same argument as in (i) shows that VD is irreducible. 1ALL LINEAlt GllOUPS Sec. 2. (1) yields that T = Z(F) = Ce(D). Let 13 = Ce(Z(D)). 5, ICB(D/Z(D))ITI divides IDIZ(D)I = IFIT!.

72 BOUNDS FOB. LINF;AR cmoups Sec. re some interesting consequences of the bounds just mentioned. ) Also, I~t pb be the order of a Sylow p-subgroup. Then SOLVABLE PERMUTATION GROUPS i) i is bounded above by a logarithmic function of b. ii) I is bounded above by a logarithmic function of r. 3 of [Wo 5J .. The bounds, which §4 Orbit Sizes of p-Groups and the Existence of , are in some sense best possible, are slightly weaker for Fermat primes. 12 (a). Let G be a permutation group ona finite set is called regular, if Gc(w) , L n.

### 4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups by Steinke G. F.

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