hw2sol - Homework 2 Solutions 1. Consider the additive...

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Homework 2 Solutions 1. Consider the additive group Z with the operation +. Which of the following are subgroups of this group? 1. { n Z : n is prime } . 2. { 0 } . 3. { 2 } . 4. { 3 n : n Z } Solution: Recall that to check if a subset of a group forms a subgroup, we must check closure, inverses, and identity. For (i) observe that 2 + 2 = 4 so closure is violated and this is not a group. For (ii) we have that { 0 } is a group. Indeed, it is the trivial group with one element. It is immediate that closure holds, that 0 is the identity, and that the inverse of 0 is 0. For (iii) we have the same contradiction as in (i), namely 2+2 = 4 so closure is violated. Finally, we observe that the set { 3 n : n Z } is a subgroup. Closure follows from the observation that the sum of two multiples of 3 is a multiple of 3, i.e. if n = 3 a and m = 3 b then n + m = 3 a + 3 b = 3( a + b ). Since 0 is a multiple of 3 we have the identity, and if m = 3 a then the inverse of m , namely - m = - 3 a = 3( - a ) is also a multiple of 3. 2. Say that a transformation f of R n is a collineation if for every line we have that f ( ) is a line. Prove that if f is a collineation, then f - 1 is also a collineation. Solution: Let R n be a line and choose two points ~x,~ y . Now, set ~x 0 = f - 1 ( ~x ) and ~ y 0 = f - 1 ( ~ y ) and let 0 R n
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hw2sol - Homework 2 Solutions 1. Consider the additive...

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