Applications of Sylow Theorems

 

Sylow Theorem's is Group Theory are an important mathematical tool. Below we give a simple application of the theorems.

Theorem: If H and K are subgroups of G and Hleq N_G(K), then HK is a subgroup of G. In particular, if K is normal is G then HKleq G for any Hleq G.

Proof: We prove HK=KH. We let hin H, k in K.
By assumption, hkh^{-1}in K, hence hk=(hkh^{-1})h in KH.
This proves that HKleq KH.
Similarly, kh=h(h^{-1}kh)in HK, proving the reverse.
We know that if H and K are subgroups of a group then HK is a subgroup if and if only HK=KH.
This proves the result.

Theorem: Let G be a group of order pq, where p and q are primes such that p<q.

  • If pmid(q-1), there exists a non-abelian group of order pq.
  • Any two non-abelian groups of order pq are isomorphic.

Proof: We let Q be a Sylow q-subgroup of the symmetric group of degree q, S_q. We know that if p is a prime and P is a subgroup of S_p of order p, then mid N_{S_p}(P) mid = p(p-1).
We know that every conjugate of P contains exactly p-1 p-cycles and computing the index of N_{S_p}(P) in S_p we can prove the above result.
So, mid N_{S_q}(Q) mid =q(q-1).
Since p mid q-1 and by Cauchy's theorem N_{S_q}(Q) has a subgroup P of order p.
Using the previous theorem we can see that PQ is a group of order pq.
Since C_{S_q}(Q)=Q so, PQ is non-abelian.
This proves the first result.

Let G be any group of order pq, let Pin Syl_p(G) and let Qin Syl_q(G).
We have pmid q-1 and let p=<y data-recalc-dims=" />.
Since Aut(Q) is cyclic, it contains a unique subgroup of order p, say <gamma data-recalc-dims=" />, and any homomorphism phi : P rightarrow Aut(Q) must map y to a power of gamma.
There are therefore p homomorphisms phi_i:P rightarrow Aut(Q) given by phi_i(y)=gamma^i, 0leq i leq p-1.
Now each phi_i for i not equal to 0 give rise to a non-abelian group G_i, of order pq.
It is straightforward to check that these groups are all isomorphic because for each phi_i,~i data-recalc-dims=0" />, there is some generator y_i of P such that phi_i(y_i)=gamma.
Thus up to a choice for the arbitary generator of P, these semi-direct products are all the same.

 

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READ:   Har Gobind Khorana
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