Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

12
ADVANCED ABSTRACT ALGEBRA
Therefore we get
Warning:
G = H + H + − − − − − − − − − + H
→ r times ←
i. e.
G = r H
Let G be a finite group of order 12. We may think that it has subgraps of order 12, 6, 4, 3, 2, 1 but no others.
Converse of Lagranges theorem is false. 6|12 but there exists a group of order 12 which does not have a
subgroup of order 6. We shall give this example some time later.
The number of right (or left) cosets of a subgroup H in a group G is called the index of a subgroup H in the
group G. This number is denoted by /G:H/. When G is finite, by Lagrange's theorem, we have G: H = G H .
We can say:
Corollary 1:
G = H × index of H in G.
a dvides G
In a finite group, the order of each element of the group divides the order of the group.
Proof:
a = 0b< a > g= order of the subgroup generated by
Corollary 2:
Groups of Prime order are cyclic.
Proof:
,
Hence the corollary.
a
Le G
Corallary3:
c h
divides
but o a ≠ 1 and G is prime. Hence o a = G.
.
c h
Therefore a ≤G G = a ie.. G is cyclic.
a |G| = e.
Proof:
let G be a finite group, and let a
G
∈ G
.Then
= a n, nis a positive integer, by Corollary 1.
G
Hence a = a
a n
n
a
e j
= a = e
= e
n