math 435 sample exam 1

math 435 sample exam 1 - Sample Questions for Midterm 1,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Sample Questions for Midterm 1, MATH 435 Problem 1 [Review the definitions and theorems covered so far. You will be asked to state a few of them precisely.] Problem 2 [Review the TRUE/FALSE questions of the relevant sections.] Problem 3 Let G be a group. An automorphism is an isomorphism f : G → G. Let Aut(G) be the set of all automorphisms of G. Show that Aut(G) with the binary operation of composition is a subgroup of S G , the symmetric group of G. Problem 4 Let A be the square below. σ 900 τ 2 1 4 3 Let σ be the symmetry of the square given by counter-clockwise rotation by 90◦ . Let τ be the symmetry given by reflection along the diagonal through nodes 2 and 4. 1. Which permutations in S 4 are represented by σ, τ, respectively? Give a representation as (products of disjoint) cycles. 2. What is the index of the subgroup H of S 4 generated by {σ, τ}? 3. Is H cyclic? Problem 5 Let G be a group. Let a ∈ G. Define the mapping fa : G → G by fa ( x) = ax for all x ∈ G. Show that fa is an element of the symmetric group S G . ...
View Full Document

Ask a homework question - tutors are online