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Lecture 2
1 Groups
A large number of sets endowed with a binary operation have properties like the set of inte- gers with addition. These systems are called groups defined as follows:
Groups: A group is a set G, together with a binary operation โ, satisfying the following properties:
- G is closed under โ, i.e for all a, b โ G, a โ b = c โ G.
- โ is associative, i.e for all a, b, c โ G, we have (a โ b) โ c = a โ (b โ c)
- G has a โ identity element i.e โe โ G such that for all a โ G a โ e = e โ a = a
- Every element in G has its โ inverse i.e for all a โ G, โb โ G such that a โ b = b โ a = e b is called the โ inverse of a, denoted as, aโ^1.
Note: Often a โ b is written as ab. This should not be confused with ordinary multiplication in numbers. Examples:
- Eg. 1 ใZ, +ใ
- Eg. 2 ใQ, +ใ
- Eg. 3 ใQโ, รใ, where Qโ^ = Q โ { 0 }
- Eg. 4 G = {a + b
2 , a, b โ Q}
ใG, +ใ is a group.
ใGโ, รใ where Gโ^ = G โ { 0 }?
Existence of (a + b
2)โ^1 if a^2 = 2b^2? Such elements are not in G. So it is a group.
- Eg. 5 ใC, +ใ and ใCโ, รใ are groups.
- Eg.6 Set of all n ร n real invertible matrices forms a group under the operation of matrix multiplication.
This group is called the general linear group of order n, denoted as GLn(R). Similarly GLn(C) is a group.
- Eg. 8 Z 4 = { 0 , 1 , 2 , 3 }. The binary operation is addition modulo 4.
a โ b = a + b(mod4).
By definition, Z 4 is closed under โ.
0 is the identity. 1 and 3 are inverses of each other. 2 is its own inverse.
For groups containing a small number of elements, a group table is a convenient way to specify the group completely.
We construct the group table of Z 4
- Eg. 9 The Klein 4 group (K 4 )
The group table of K 4 = {e, a, b, c} is
e a b c e e a b c a a e c b b b c e a c c b a e
2 Subgroups
Def.Subgroup: Let ใG, โใ be a group. A non-empty subset H of G is called a subgroup of G if ใH, โใ is a group.
- 2 Z = {...., โ 6 , โ 4 , โ 2 , 0 , 2 , 4 , 6 , ....} = { 2 k|k โ Z} ใ 2 Z, +ใ is a subgroup of ใZ, +ใ
- ใZ, +ใ is a subgroup of ใR, +ใ is a subgroup of ใC, +ใ.
- Let M be the set of real 2 ร 2 matrices with determinant =1. Then M is a subgroup of GL 2 (R).
- Lemma 3: A non-empty subset H of a group ใG, โใ is a subgroup of G if and only if (i) H is closed under โ. (ii) a โ H =โ aโ^1 โ H.
Eg: Let n โ Z and consider the set nZ.
Let nk 1 , nk 2 โ nZ where k 1 , k 2 โ Z.
Then nk 1 + nk 2 = n(k 1 + k 2 ) โ nZ since Z is closed under addition.
So nZ is closed under addition.
For any nk โ nZ, n(โk) โ nZ, which is its additive inverse.
So by Lemma 3 ใnZ, +ใ is a subgroup of ใZ, +ใ.
- Lemma 4: If H is a non-empty finite subset of a group ใG, โใ, and H is closed under โ then H is a subgroup of G.
Proof: Since H is non-empty, โa โ H. Since H is closed under โ, a, a^2 , ..... โ H.
But H is finite. So โr, p โ Z, p > r such that ap^ = ar^ =โ apโr^ = e โ H.
So e โ H.
Now a(pโr)โ^1 โ a = a โ a(pโr)โ^1 = apโr^ = e.
So a(pโr)โ^1 = aโ^1.
Hence โ a โ H, aโ^1 โ H. By Lemma 3, H is a subgroup of G.
