5. True or False. Explain your answers. (a) If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order. (b) If |G/H|= =p, then the order of every element a H in G is a multiple of p.

Find all values of k so that k 2 3 k+1 is invertible. -1₂ To enter your answer, first select 'always', 'never', or whether k should be equal or not equal to specific values, then enter a value or a list of values separated by commas. Invertible: Always

Let A = {a, b, c, d, e, f, g}, B = {b, a, g}, C = {f, a, d, e} and D ={a, g, e, d}.(i) What is the powerset of B?(ii) Let n = |P(C)| and m = |B × D|. What are n and m?(iii) Draw an Euler diagram of A, B and C.(iv) Draw a Venn diagram of A, B and C.(v) Let E = P (A \ C) ∩ (A \ (D \ B))
. What is |E|

Suppose a permutation s swaps the first two elements of {1, 2, ..., n}. A permutation r rotates the elements: r(1)=2, r(2)=3, ... r(n-1)=n, r(n)=1. We can thus write in cycle notation: s=(1 2), r=(1 2 3 4 ... n). What does the permutation rm sr -m do?