A 2-Sylow subgroup of J is elementary abelian of order 8 and J has no subgroup of index 2. If r is an involution in J, then C(r) = (r) X K, where K _ A5. Let G be a finite group with the following properties: (a) S2-subgroups of G are abelian; (b) G has no subgroup of index 2; and (c) G contains an involution t such that 0(t) = (t) X F, where F A5. Then G is a (new) simple group isomorphic to J...

G. Navarro raised the question whether the ordinary character table X(G) of a finite group G determines the Sylow numbers of G. In this note we show that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are determined by their ordinary character table. If G and H are finite groups with i...

Solution. The prime factorization of 185 is 5*37. Given a group of order 185, Let n5 be the number of subgroups of order 5 and n37 the number of subgroups of order 37. Since subgroups of order 5 and 37 will be Sylow subgroups, by the Sylow Theorems n5 and n37 have to be 1 (mod 5) and 1 (mod 37), respectively and divide 37 and 5, respectively. Hence, n5 = n37 = 1 and the 5 and 37 Sylow subgroups...

THEOREM. Let G be a finite group and s an involution in G, such that CG(S)~CGQ(SO)Assume G^Co(s)0(G) (0(G) is the maximal normal odd order subgroup of G). Then Gc^Go. In particular, G has the following properties : (i) G has order 2 • 3 • 5 • 7 • 11 • 23, and is simple. (ii) G has two conjugacy classes of involutions. One class is represented by the involution s. A representative t of the secon...

In the Sylow theorems f we learn that if the order of a group 2Í is divisible hj pa (p a prime integer) and not by jo*+1, then 31 contains one and only one set of conjugate subgroups of order pa, and any subgroup of 21 whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of 2Í. These conjugate subgroups may be called the Sylow subgroups of 21. It will be o...

The author has computed the class groups of all complex quadratic number fields Q(\f^~D) °f discriminant D for 0 < D < 4000000. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are D = 3321607 = prime, class group C(3) x C(3) x C(9.7) (C(n) a cyclic group of order n), and D = 3640387 = 421.8647, class group C(3) X C(3) X C(9.2). The author has also foun...

A complete Sylow sequence, P = P1, . . . , Pm , of a finite group G is a sequence of m Sylow pi -subgroups of G, one for each pi , where p1, . . . , pm are all of the distinct prime divisors of |G|. A product of the form P1 · · · Pm is called a complete Sylow product of G. We prove that the solvable radical of G equals the intersection of all complete Sylow products of G if, for every compositi...

In this note we introduce the notion of a group action on a set and use it to prove a number of theorems about p-groups and the Sylow theorems. For undefined terms see any decent book on group theory. The theory of p-groups and the Sylow theorems have a number of applications in Galoistheory. The nice structure of p-groups will translate via the fundamental theorem of Galois theory to nice stru...

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

Originally this was to be entirely on Geometry in computer algebra systems|especially CAYLEY/MAGMA and GAP (but perhaps also at least partially in other systems such as MAPLE)|to some extent just a wish list. However, at the request of John Cannon, that will be the second of two parts: I'll start with PART I. Some geometric algorithms (and applications). There are a couple of ways to phrase the...