hw6.pdf - MODERN ALGEBRA 1 HOMEWORK 6(1 Chapter 2 8.3(2...

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MODERN ALGEBRA 1: HOMEWORK 6 (1) Chapter 2: 8.3 (2) Chapter 2: 8.6 (3) Chapter 2: 12.5 (4) Let H G be a subgroup. The normalizer of H , denoted by N ( H ) , is defined by N ( H ) = { g G | gHg - 1 = H } . (a) Show that N ( H ) is a subgroup of G and N ( H ) = G if and only if H / G . (b) Show that H is a normal subgroup of N ( H ) . Remark. Keep in mind that saying gHg - 1 = H is not the same as saying ghg - 1 = h for every h H . By gHg - 1 = H we mean that the collection { ghg - 1 | h H } as a whole is the same as the collection { h | h H } . (5) Let G = GL 2 ( R ) and H G the subgroup of diagonal matrices, namely matrices of the form a 0 0 b , ab 6 = 0. Find N ( H ) and identify the quotient group N ( H ) / H . (Optional) Generalize the previous result to GL n ( R ) . (6) Let Q =
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