Understanding Subgroups in Group Theory

1. SECTION 5 Subgroups Notation and TerminologyIn general , we shall use “e” to denote the identity element of a group Multiplicative Group Additive Group

2. Subgroups Definition: If G is a group, then the order |G| of G is the number of elements in G. Definition: If a subset H of a group G is closed under the binary operation of G and if H with the induced operation from G is itself a group, then H is a subgroup of G. We shall let H G denote that H is a subgroup of G, and H < G denote HG but H G. Note : Every group G has G and {e} as its subgroups.

3. Examples Definition: If G is a group, then the subgroups consisting of G itself is the improper subgroup of G. All other subgroups are proper subgroups. The subgroup {e} is the trivial subgroup of G. All other subgroups are nontrivial. Example: •  Z, + <  R, +  but  Q+,  is not a subgroup of  R, + . •  Q+, <  R+, 

4. Group Zn When a=qn+r, where q is the quotient and r is the remainder upon dividing a by n, we write a mod n = r . Example: 12 mod 4 = 0, 3 mod 4 = 3 etc. • The set Zn={0, 1, …, n-1} for n 1 is a group under addition modulo n. This group is referred as the group of integers modulo n. Example: Z4={0, 1, 2, 3} under addition modulo 4 + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2

5. Klein 4-group The set V={e, a, b, c} under the following operation is a group. We call it the Klein 4-group. e a b c e e a b c a a e c b b b c e a c c b a e

6. Subgroup Diagrams of Z4 and V • The only nontrivial proper subgroup of Z4 is {0, 2}. Here {0, 3} is not a subgroup since it’s not closed under +. • However, the group V has three nontrivial proper subgroups: {e, a}, {e, b} and {e, c}. Here {e, a, b} is not a subgroup since it’s not closed under the operation of V. Z4 V {0, 2} {e, a} {e, b} {e, c} {0} {e}

7. Theorem 5.14 A subset H of a group G is a subgroup of G if and only if • H is closed under the binary operation of G, • The identity element e of G is in H, • For all a  H it is true that a1  H also.

8. Cyclic Subgroups Theorem Let G be a group and let a  G. Then H={ an | n  Z} is a subgroup of G and is the smallest subgroup G that contains a, that is, every subgroup containing a contains H. Proof: We check the three conditions given in Theorem 5. 14. • Since ar as=ar+s for r, s  Z, we see that H is closed under the group operation of G. • Also a0=e, so e H, • for ar  H, a-r  H and ar a-r =e. Hence H G. Since every subgroup of G containing a must contain all element an for all n  Z, H is the smallest subgroup of G containing a.

9. Generator Definition: Let G be a group and let a  G. Then the subgroup { an | n  Z} of G, characterized in Theorem 5.17, is called the cyclic subgroup of G generated by a, and denoted by  a . An element a of a group G generates G and is a generator for G if  a =G. A group is cyclic if there is some element a in G that generates G. Example: • Z4 is cyclic and  1 =  3 = Z4 • V is not cyclic since  a,  b ,  c  are subgroups of two elements, and  e  is the trivial subgroup of one element.

10. Examples •  Z, +  is a cyclic group:  1  = -1  =Z, and no other generators • For n  Z+, Zn under addition modulo n is cyclic. If n>1, then  1  =  n-1 = Zn, there may be others. • nZ=  n  is a cyclic subgroup generated by n consists of all multiples of n, positive, negative, and zero. Note that 6Z < 3Z.