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 Post subject: Subgroups of Complex NumbersPosted: Sun, 2 Nov 2003 00:28:13 UTC
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Joined: Sun, 28 Sep 2003 21:18:37 UTC
Posts: 69
1. U = {z is an element of C such that the absolute value of z = 1)

Show that U is a subgroup of (C, *).

2. Un = {z is an element of C such that z^n = 1}

Show that for each positive integer n >= 2, Un is a subgroup of (C, *). To what groups is Un isomorphic?

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 Post subject: Re: Subgroups of Complex NumbersPosted: Sun, 2 Nov 2003 08:43:42 UTC
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Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9961
Mathman wrote:
1. U = {z is an element of C such that the absolute value of z = 1)

Show that U is a subgroup of (C, *).

To prove that a group H is a subgroup of another group G under multiplication, you need to show the following:

1) H is closed under muliplication
2) The identity element e of G is contained in H
3) If is an element of H, then so is a^{-1}

Mathman wrote:
...To what groups is Un isomorphic?

My suggestion would be to multiply arbitrary elements of U_n together and see what you get. Start with small examples, such as U_4.

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