Sylow theory.?

Question

What is Sylow theory?(in groups)

Answer 1

First Sylow Theorem:  Let G be a finite group.  If p is a prime such that p^alpha is a divisor of | G | for some alpha >= 0, the G contains a subgroup of order p^alpha.

Answer 2

Fasten your seatbelt and open your mind.  This is heavy stuff.

Let p be a prime number; then we define a Sylow p-subgroup (sometimes p-Sylow subgroup) of G to be a maximal p-subgroup of G (i.e., a subgroup which is a p-group, and which is not a proper subgroup of any other p-subgroup of G). The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G).

Collections of subgroups which are each maximal in one sense or another are not uncommon in group theory. The surprising result here is that in the case of Sylp(G), all members are actually isomorphic to each other, and this property can be exploited to determine other properties of G.

In particular, if |G| = pnm and gcd(p, m) = 1 then any Sylow p-subgroup P has order |P| = pn. That is, P is a p-group and gcd(|G:P|, p) = 1. Thus if gcd(|G|, p) = 1 then P is a Sylow p-subgroup iff P = {1}.

The following theorems were first proposed and proven by Norwegian mathematician Ludwig Sylow in 1872, and published in Mathematische Annalen. Given a finite group G and a prime p which divides the order of G, we can write the order of G as pn · s, where n > 0 and p does not divide s.

Theorem 1: There exists a Sylow p-subgroup of G, of order pn.

The following weaker version of theorem 1 was first proved by Cauchy.

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element of order p in G .

Theorem 2: If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g−1Hg is a subgroup of P. In particular, all Sylow p-subgroups of G are conjugate (and therefore isomorphic) to each other, i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K.

Theorem 3: Let np be the number of Sylow p-subgroups of G.

    * np divides s.

    * np = 1 mod p.

    * np = |G : NG(P)|, where P is any Sylow p-subgroup of G and NG denotes the normalizer.

In particular, the above implies that every Sylow p-subgroup is of the same order, pn; conversely, if a subgroup has order pn, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pn.

A very important consequence of Theorem 3 is that the condition np = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup. (There are groups which have normal subgroups but no normal Sylow subgroups, such as S4.)

There is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in an infinite group to be a p-subgroup (that is, every element in it has p-power order) which is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorn's lemma.

Theorem: If K is a Sylow p-subgroup of G, and np = |Cl(K)| is finite, then every Sylow p-subgroup is conjugate to K, and np = 1 mod p, where Cl(K) denotes the conjugacy class of K.