Sylow Theorems Flashcards
Conjugates?
Let a,b in G. a and b are conjugate in G if xax^{-1}=b for some x in G.
Conjugacy class of a?
cl(a) = {xax^{-1} | x in G} i.e. the set of all conjugates of a.
Center of a group?
Z(G) = {a in G | ax=xa for all x in G} i.e. the set of all elements that commute with every element in G.
Centralizer of a?
C(a) = {g in G | ga=ag} i.e. the set of all elements in G that commute with a.
If G is a finite group, what is the number of conjugates of a in G?
|cl(a)| = |G:C(a)|
What is the index of H in G? |G:H|?
The number of distinct left cosets of H in G.
|G:H| = |G|/|H|
What is the class equation of a finite group G?
|G| = \sum |G:C(a)| where the sum runs over one element a from each conjugacy class of G.
If |G| = p or |G|=p^2, then?
G is Abelian.
Sylow’s First Theorem. Existence Theorem.
If G is a finite group, and p be prime. If p^k divides |G|, then G has at least one subgroup of order p^k.
What is a Sylow p-subgroup?
If p^k divides |G|, but p^{k+1} does not, then any subgroup of G of order p^k is a Sylow p-subgroup.
Let G be a finite group and p a prime that divides |G|. Then?
G has an element of order p.
Sylow’s second Theorem.
If H is a subgroup of finite group G and |H| is a power of p, then H is contained in some Sylow p-subgroup of G.
Sylow’s Third Theorem.
Let p be prime and |G| = p^km where p does not divide m. Then the number of Sylow p-subgroups of G is equal to 1 modulo p and divides m.
Furthermore, any two Sylow p-subgroups are conjugate.
When is a Sylow p-subgroup normal?
If and only if it is the only Sylow p-subgroup.
