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thandu3

Member

Registered: 2008-10-21

Posts: 4

normal and quotient subgroup problem

1)    Z   is a normal subgroup of R(real number under addition.).Show that the quotient group R/Z is isomorphic to the circle group K.
2)    Prove that every quotient group of a cyclic group is cyclic.
3)    Let H is a normal subgroup of a group G. Show that the order of aH as an element of the quotient group G/H divides the order of a єG.
can anyone help me doing these?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: normal and quotient subgroup problem

1.  Any time you are proving G/K is isomorphic to H, construct a map from G onto H with kernel K, and use the fundamental homomorphism theorem (also know as the 1st isomorphism theorem).

2. As with all proofs concerning cyclic groups, start with stating there exists an element g such that <g> = G.  Now what happens when you use powers of g to represent cosets?

3. Suppose a has order n.  What is (aH)^n?  You should also think about the question, "Why is it divides the order of a, and not equal to it?"

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