MATH 417 Study Guide - Midterm Guide: Complex Multiplication, Integral Domain
16 Oct 2018
School
Department
Course
Professor
Department of Mathematics
University of Illinois at Urbana-Champaign
MATH 417 – SPRING 2017 – SECTION B1
FEBRUARY 10, 2017
Midterm Duration: 50 m
SOLUTIONS
1. (25 points) Consider the group H={−1,1}with the usual
multiplication of integers. Show that every element gin the
group G=H×Hsatisfies g2= 1.
Solution: The elements of Gare pairs g= (h1, h2) with
h1, h2∈H. The identity in Gis 1 = (1,1) and multiplication
is component by component, hence:
g2= (h1, h2)(h1, h2) = (h2
1, h2
2) = (1,1) = 1.
1
2 MATH 417 – 1ST MIDTERM
2. (25 points) Consider the permutation σ∈S7given by:
σ=1 2 3 4 5 6 7
7 6 4 1 2 5 3
i) Find a decomposition of σinto a product of disjoint cycles.
ii) Find two distinct decompositions of σinto a product of
transpositions.
Solution:
i) σ= (1,7,3,4)(2,6,5).
ii) σ= (1,4)(1,3)(1,7)(2,5)(2,6) = (7,1)(7,4)(7,3)(2,5)(2,6).
Document Summary
Math 417 spring 2017 section b1. Solutions: (25 points) consider the group h = { 1, 1} with the usual multiplication of integers. Show that every element g in the group g = h h satis es g2 = 1. Solution: the elements of g are pairs g = (h1, h2) with h1, h2 h. the identity in g is 1 = (1, 1) and multiplication is component by component, hence: g2 = (h1, h2)(h1, h2) = (h2. 1, h2: = (1, 1) = 1. 2: (25 points) consider the permutation s7 given by: Math 417 1st midterm (cid:18) 1 2 3 4 5 6 7 (cid:19) 7 6 4 1 2 5 3: find a decomposition of into a product of disjoint cycles, find two distinct decompositions of into a product of transpositions. Solution: = (1, 7, 3, 4)(2, 6, 5), = (1, 4)(1, 3)(1, 7)(2, 5)(2, 6) = (7, 1)(7, 4)(7, 3)(2, 5)(2, 6).
