# Math Normal Group

1. If H and K are subsets of a group define HK={ hk | h Ԑ H and k Ԑ K }

(a) Prove that if H is a normal subgroup of a group, then HaHb=Hab, i.e the coset.

Hence, if H is a normal subgroup of a group, the operation *, defined by Ha*Hb=Hab is a well defined operation on the set of right coset of H in G.

(b) Prove that { Ha | a Ԑ G} together with the operation * forms a group. This group is denoted by G/H and is called the qoutient group of G modulo H. What is the identity element? The inverse of Ha?

2. In parts (a) - (c), all the qoutient groups are cyclic and therefore are isomorphic to Zn for some n. In each case, find n, and a generator of the qoutient group.

(a) Z6/<>

(b) Z12/<>

(c)Z15/<>

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1 (a)

HaHb=HHab as H is normal

Now HH ={ hk | h inH and k in H} = H

Therefore

HaHb=Hab

(b) from (a) HaHb=Hab is a right coset of H in G.

H = He is the identity HHa = Ha for all a in G

Ha^-1 is the inverse of Ha

Hence G/H is a group.

(2)

(a) n=3 generator 1+ 

(b) As  =  in Z12 n =4 generator 1+

(c) As  =  in Z15. n=3 generator 1+

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