Lecture 2
1 Lagrange’s Theorem
Consider Z12
h3i = {0, 3, 6, 9}
1 6∈ h3i
Let us add 1 to each element of h3i and denote this set as 1h3i
1h3i = {1, 4, 7, 10}
2 6∈ h3i ∪ 1h3i
2h3i = {2, 5, 8, 11}
We observe that the sets h3i, 1h3i and 2h3i are disjoint.
and h3i ∪ 1h3i ∪ 2h3i = Z12 .
All the 3 sets above have 4 elements.
⇒ o(h3i) divides o(Z12 ).
Let G be a finite group and H be a proper subgroup of G.
Consider aH where a 6∈ H.
Let x ∈ H ∩ aH, then x = ah1 = h2 for some h1 , h2 ∈ H.
⇒ a = h2 h−1 1 ∈ H; ⇒⇐
⇒ H ∩ aH = ∅
In the set aH, ah1 = ah2 ⇒ h1 = h2 ⇒ |aH| = |H|.
Consider b ∈ G such that b 6∈ H and b 6∈ aH
Then H ∩ bH = ∅.
Let x ∈ aH ∩ bH ⇒ x = ah1 = bh2 for some h1 , h2 ∈ H.
⇒ b = ah1 h−12 , ⇒ b ∈ aH ⇒⇐
So aH ∩ bH = ∅.
Since G is finite we stop when we are left with no elements of G
G has been partitioned into a collection of disjoint subsets of G all of equal size, that of H.
This construction proves what we state as :
• Lagrange’s Theorem:
If G is a finite group and H is a subgroup of G then o(H)|o(G).
• Corollary 1:
Let a ∈ G. Then ao(G) = e.
• Corollary 2:
If o(G) = p a prime then G is a cyclic group.
Cosets
Let H be a subgroup of a group G.
Consider the collection of sets {aH| a ∈ G}.
We observed the following:
• Not all the sets in this collection are distinct.
• Either aH = bH or aH ∩ bH = ∅ for a, b ∈ G and
• G = ∪a∈G aH
Def.
Left coset: Let H be a subgroup of G and a ∈ G. A left coset of H in G is a subset aH of G given
by aH = {ah|h ∈ H}.
We can similarly define the right coset as the set Ha.
Note: For a non-abellian group we may not have aH = Ha.
• Eg. 2 Multiplication ( mod 7)
Z∗7 = {1, 2, 3, 4, 5, 6}
H = {1, 6} is a subgroup of Z∗7 .
• The distinct left cosets of H are
H = 6H = {1, 6}, 2H = 5H = {2, 5}, 3H = 4H = {3, 4}
• Since this group is abellian the left cosets are same as the respective right cosets.
Cosets establishes an equivalence relation amongst the elements of the group.
• Let us define a relation on G.
a ∼ b if a ∈ bH.
• We can show that this is an equivalence relation. exercise
• The equivalence classes are the cosets of H in G.
If x ∈ [a] then x ∈ aH, ⇒ [a] ⊆ aH
If x ∈ aH then x ∼ a, ⇒ x ∈ [a] ⇒ aH ⊆ [a]
=⇒ [a] = aH.
2
• Note that the right cosets induce a different equivalence relation on G.
The equivalence classes induced by it are the right cosets and they are in general different
from the left cosets.
• Def.Index
Let H be a subgroup of a group G. Index of H in G denoted as [G : H] is defined as the
number of distinct left(or right) cosets of H in G.
• If G is finite then
o(G)
[G : H] =
o(H)
Index of H in G is also denoted as iG (H).
