Question

if H is a subgroup of a group G and if a ,b belongs to G the​

Answer 1

Step-by-step explanation:

Definition: A subset H of a group G is a subgroup of G if H is itself a group under the

operation in G.

Note: Every group G has at least two subgroups: G itself and the subgroup {e},

containing only the identity element. All other subgroups are said to be proper

subgroups.

Examples

1. GL(n,R), the set of invertible

†

n ¥ n matrices with real entries is a group under matrix

multiplication. We denote by SL(n,R) the set of

†

n ¥ n matrices with real entries whose

determinant is equal to 1. SL(n,R) is a proper subgroup of GL(n,R) . (GL(n,R), is called

the general linear group and SL(n,R) the special linear group.)

2. In the group

†

D4 , the group of symmetries of the square, the subset

†

{e,r,r

2

,r

3

} forms a

proper subgroup, where r is the transformation defined by rotating

†

p

2

units about the z-

axis.

3. In

†

Z9 under the operation +, the subset {0, 3, 6} forms a proper subgroup.

Problem 1: Find two different proper subgroups of

†

S3.

We will prove the following two theorems in class:

Theorem: Let H be a nonempty subset of a group G. H is a subgroup of G iff

(i) H is closed under the operation in G and

(ii) every element in H has an inverse in H.

For finite subsets, the situation is even simpler:

Theorem: Let H be a nonempty finite subset of a group G. H is a subgroup of G iff H is

closed under the operation in G .

Problem 2: Let H and K be subgroups of a group G.

(a) Prove that

†

H «K is a subgroup of G.

(b) Show that

†

H »K need not be a subgroup

Example: Let Z be the group of integers under addition. Define

†

Hn to be the set of all

multiples of n. It is easy to check that

†

Hn is a subgroup of Z. Can you identify the