Page 54 - DMTH403_ABSTRACT_ALGEBRA

Unit 3: Subgroups

Now we state a corollary to Theorem 8, in which we write down the important point made in the Notes
proof of Theorem 8.
Corollary: Let H  {e} be a subgroup of < a >. Then H = < a >, where n is the least positive integer
n
such that a  H.
n
Self Assessment

1. Subgroup of an abelian group is ..................

(a) normal (b) abelian
(c) cyclic (d) homomorphism
2. If H be a non-empty subset of a group G. Then H is a subgroup of G if a, b  H, then

(a) ab  H (b) a b  H
-1
-1
(c) ab  H (d) a  H, b  H
-1
–1
3. .................. is a Greek letter omega.
(a)  (b) 
(c)  (d) 

4. Let G be a group, H be subgroup of G and be subgroup of H then k is a .................. of G.
(a) normal group (b) cyclic group
(c) abelian group (d) subgroup

5. Let H and k be subgroups of a group G. Then KH = HK then (HK) = ..................
–1
(a) k , h  Hk (b) h k  Hk
-1
k
-1
h
(c) x/h  Hk (d) H/k  Hk
3.4 Summary
In this unit we have covered the following points.
Here we discussed the definition and examples of subgroups.

The intersection of subgroups is a subgroup.

The union of two subgroups H and K is a subgroup if and only if H  K or K  H.

The product of two subgroups H and K is a subgroup if md only if HK = KH.

The definition of a generating set.

A cyclic group is abelian, but the converse need not be true.

Any subgroup of a cyclic group is cyclic, but the converse need not be true.

3.5 Keywords

Subgroup: Let (G,*) be a group. A non-empty subset H of G is called a subgroup of G if

(i) a * b  H  a, b  H, i.e., * is a binary operation on H,
(ii) (H,*) is itself a group.

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