s09_mthsc851_hw08

# s09_mthsc851_hw08 - MTHSC 851(Abstract Algebra Dr Matthew...

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Unformatted text preview: MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 8 Due Thursday April 2nd, 2009 (1) Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements is a homomorphic image of N (i.e., free objects do not always exist in the category C of nilpotent groups). (2) Let G be a group with X ⊆ G , and let A be the normal subgroup generated by X , i.e., A = { N C G : X ⊆ N } . Let Y = { gxg- 1 | x ∈ X, g ∈ G } . Show that A = h Y i . (3) Show that the Klein 4-group V has presentation h a,b | a 2 = b 2 = ( ab ) 2 = 1 i . (4) Show that the quaternion group Q 2 = {± 1 , ± i, ± j, ± k } has presentation h a,b | a 4 = 1 , a 2 = b 2 , ab = ba 3 i . (5) (a) Determine the group with presentation h a,b | a 2 = 1 , b 3 = 1 , ab = ba i . (b) If G and H each have more than one element, show that G * H is an infinite group with center h e i ....
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