Let the symmetric group Se act on the set X = {1,...,4} in the usual way:...
Let n EN Consider the set of n x n symmetric matrices over R with the usual addition and multiplication by a scalar (1.1) Show that this set with the given operations is a vector subspace of Man (6) (12) What is the dimension of this vector subspace? (1.3) Find a basis for the vector space of 2 x 2 symmetric matrices (6) (16)
Question 4
Exercise 1. Let G be a group such that |G| is even. Show that there exists an EG,17e with x = e. Exercise 2. Let G be a group and H a subgroup of G. Define a set K by K = {z € G war- € H for all a € H}. Show that (i) K <G (ii) H <K Exercise 3. Let S be the set R\ {0,1}. Define functions from S to S by e(z)...
4. Let M be the set of 2 x 2 matrices of the form (62) where a, d E R - {0}. Consider the usual matrix multiplication ·, i.e: ae + bg af + bh ce + dg cf + dh (a) Show that (M,·) is an abelian group. 1 (b) Compute the cyclic subgroup generated by M = What is the order of M? 66 -4) (1) EM EM.
31. Let Sa be the symmetric group on three letters. Describe, as given in class, the embedding of S3 into Se through any ordering of the generators you choose.
31. Let Sa be the symmetric group on three letters. Describe, as given in class, the embedding of S3 into Se through any ordering of the generators you choose.
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair? Use Corollary 1.4 to recompute |Tetral. 1. Corollary 1.4. If a finite group G acts on a set S, then for any SES, IG| = #(Os)|G5|. In particular, # (0s) divides |G|.
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair?...
Consider the rotational symmetry group G of the cube Let X be the set of edges of the cube, and let xe X be the edge between faces A and E (see picture). G acts on X in the obvious way. Describe the stabilizer Stabg(x) and the orbit Orbg(x). By using the orbit-stabilizer theorem, deduce G.
4) Let G be a group and let a є G. The centralizer of a in G is defined as the set (i) Show that Ca(a) is a subgroup of G (ii) Find the centralizers of the elements r and y in the Dihedral group D4
Let M be the set of 2 x 2 matrices of the form (82) where a, d ER-{0}. Consider the usual matrix multiplication, i.e: ae + bg af +bh ce + dg cf + dh (2)) = (ce ) (a) Show that (M,-) is an abelian group. (b) Compute the cyclic subgroup generated by M = What is the order of M? (6 -4) € M.
(4) Write the quaternion group in the usual fashion as Qs = {1,-1, i, -i, j, - j, k, -k) and consider the subgroup H = {1,-1). Write down the multiplication table for Qs/H.
Problem 4. Let G be a group. Recall that the order of an element g G is the smallest k such that gk = 1 (or 00, if such a k doesn't exist). (a) Find the order of each element of the symmetric group S (b) Let σ-(135)(24) and τ-(15)(23)(4) be permutations in S5. Find the cycle decompositions for (c) Let σ-(123456789). Compute ơ-i, σ3, σ-50, and σί006 (d) Find all numbers n such that Ss contains an element of...