promotion image of download ymail app
Promoted
Sandy 發問於 科學及數學數學 · 9 年前

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/<[2]>

(b) Z12/<[8]>

(c)Z15/<[6]>

1 個解答

評分
  • 9 年前
    最愛解答

    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+ [2]

    (b) As [8] = [4] in Z12 n =4 generator 1+[8]

    (c) As [6] = [3] in Z15. n=3 generator 1+[3]

    • Commenter avatar登入以回覆解答
還有問題嗎?立即提問即可得到解答。