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HW 1

February 5, 2018

Due Wed, Feb 21. Be sure to justify all of your answers to receive full credit. In this assignment we will show that a group of order 6 is either commutative, or is isomorphic

to S3.

1. Let G be a group. Suppose that for any element x ∈ G, x2 = e. Show that G is abelian.

2. Suppose G has even order, show that there is at least one element of G which has order 2.

3. Now let |G| = n. Suppose x ∈ G has order o(x) > n 2 . Show that o(x) = n.

4. Now let G be a group of order 6 such that G is not abelian.

(a) Show that G has an element σ of order 3, and an element τ of order 2.

(b) Show that στ = τσ2.

(c) Conclude that G is isomorphic to S3.

5. Saracino: 2.10, 3.1, 3.4, 3.6, 4.6, 4.13, 5.4, 5.6, 5.25

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