solution10 - PMath 336: Introduction to Group Theory...

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Unformatted text preview: PMath 336: Introduction to Group Theory Exercise set 10 July 25, 2008 Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works. Notation GL n ( S ) : Invertible n n matrices with coefficients in S , under multiplication D n : The group of symmetries of the regular n-gon S n : The group of permutations of 1 ,...,n Z : The group of integers under addition Z n : The group { ,...,n- 1 } with addition modulo n U n : The group of numbers less than n and prime to n , with multiplication mod n . Q (quaternions): The subgroup of GL 2 ( C ) generated by i = i i and j = 0 1- 1 0 . Questions 1. For each of the following maps m : G X- X , determined whether it is a group action. (a) G = D 4 , X = Z , m ( g,n ) = n + | g | - 1. (b) G = S 3 , X = Z , m ( s,n ) = 3 t + s ( r ), where r is the residue of n mod 3, n = 3 t + r , and s is viewed as a permutation of the set { , 1 , 2 } . (c) G = Q * , X = R , m ( g,x ) = gx . (d) G = Q * , X = Q , m ( g,x ) = g + x . (e) G = Q , X = Q * , m ( g,x ) = g + x . (f) G is any group, X = G , m ( g,h ) = g- 1 hg . Solution: (a) No, if g is not the identity, the action of g is not the inverse of the action of g- 1 . (b) Yes, this action is induced by the usual action of S 3 on 0 , 1 , 2. (c) Yes, since multiplication is associative (d) No, since x + gh is not equal to x + g + h (e) This is not even a well-defined function: g + x might be 0. (f) No, since in general, ( g 1 g 2 )- 1 hg 1 g 2 = m ( g 1 g 2 ,h ) is not equal to g- 1 1 g- 1 2 hg 2 g 1 = m ( g 1 ,m ( g 2 ,h )). However, this defines an action of the opposite group G . 2. Let m : G X- X be a group action. Prove the facts mentioned in remark 137, namely: (a) If Y X , the subsets G Y = { g G | gy = y for all y Y } and G [ Y ] = { g G | g ( Y ) = Y } Are subgroups that act on Y and on X- Y ....
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This note was uploaded on 04/27/2010 for the course PMATH 336 taught by Professor Moshekamensky during the Spring '08 term at Waterloo.

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solution10 - PMath 336: Introduction to Group Theory...

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