Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, ...
10. Let G = D. be the dihedral group on the octagon and let N = (r) be the subgroup of G generated by r4. (a) Prove that N is a normal subgroup of G. (b) If G =D3/N, find G. (c) Using the bar notation for cosets, show that G = {e, F, 2, 3, 5, 87, 82, 83}. Hint: Show that the RHS consists of distinct elements and then use part (b). (d) Prove that G-D4. Hint: It...
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...
Let G be a group of order 231 = 3 · 7 · 11. Let H, K and N denote sylow 3,7 and 11-subgroups of G, respectively. a) Prove that K, N are both proper subsets of G. b) Prove that G = HKN. c) Prove that N ≤ Z(G). (you may find below problem useful). a): <|/ is a normal subgroup, i.e. K,N are normal subgroups of G (below problem): Let G be a group, with H ≤ G...
Exercise 4.5.3. Let G-(g g 1 be a group of order 2 and V a CG-module of Let u +202 +2,u2 2v1 - 2 +2vs,u vector space spanned by ui, for i-1,2,3 2v - 202 +vs, and hence U the (i) Prove that U is a CG-submodule of V fori 1,2,3, and that (ii) Let λ C and u-ul + U2 + λν3 V. Find the value(s) of λ for which the subspace U spanned by u is a CG-submodule...
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
Always give rigorous arguments I. (A) Let G be a group under * and let g E G with o(g) = n (finite) (i) Show that g can never go back to any previous positive power of g* (1k< n) when taking up to the nth power (cf. g), e., that there are no integers k and m such that 1< k<m<n and such that g*-gm (ii) How many elements of the set (e, g,g2.... .g"-) are actually distinct? (iii)...
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)...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
Answer each question in the space below. 1. Let A = {0,1} U... U{0,1}5 and let be the order on A defined by (s, t) €< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element & is minimal if there does not erist Y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give...
Q16 S CX}. If G has a group 15. The powerset of a set, X, is defined to be the collection of all subsets of X: P(X) = { S action on X, then the group action can be defined on P(X) by a. S = {a.s | SES}. (a) Show that if S = orb(r), then a.S= S for all a E G. (b) If a. S = S, show that S = U; orb(r) for some elements r;...