# Math 545 midterm 1

 Definition 1.4 A nonempty set G equipped with an operation * on it is said to form a group under that operation if the operation obeys the following laws, called the group axioms: 1. Closure: for any a,b E G, we have a*b E G2. Associativity: For any  a, b, c E G, we have a*(b*c)=(a*b)*c3. Identity: There exists an element e E G such that for all a E G we have a*e=e*a=a. Such an element  e E G is called an Identity in G.4. Inverse: For each a E G there exists an element a^1 E G such that a*a^-1=a^-1*a=e. Such an element a^-1 E G is called an inverse of a in G. Proposition 1.18 Basic group properties: For any group G: 1. The identity element of G is unique2. For each a E G, the inverse a^-1 is unique3. For any a E G, (a^-1)^-1=a4. For any a, b E G, (ab)^-1=b^-1a^-15. for any a, b E G, the equations ax=b and ya=b have unique solutions or, in other words, the left and right cancellation laws hold Theorem 1: Subgroup test A nonempty subset H of a group G is a subgroup of G IFF the following condition holds: For every a, b E H, ab^-1 E HFor addition      For every a, b E H, a-b E H Theorem 2: Subgroup test 2 A nonempty subset H of a group G is a subgroup of G IFF the following conditions hold: 1. Closure: For every a, b E H, ab E H2. inverse: For every b E H, b^-1 E H Definition 2.13  Let G be a group and a E G. Then we define ={a^n|n E Z}If the group operation is written as addition, ={na| n E Z} Definition 2.19 Let G be a group and a E G. The order |a| of a in G is? the least positive integer n such that a^n=e, of infinitie if there is no such n. If the group operation is written as addition, the condition would be written na=0 Definition 2.23 Let G be any group, Then the center of G, denoted Z(G), consists of? The elements of G that commute with every element of G. In other words:            Z(G)={x E G | xy=yx, Vy E G}Note: ey=y=ye for all y E G, so E E Z(G), and the center is a nonempty subset of G. Theorem 2.24: The Center Z(G) of a group is? a subgroup of G. Definition 2.26 Let G be a group and a E G. Then the centralizer of a in G, denoted by CG(a), is? The set of all elements of G that commute with a. In other words:               CG(a)={y E G | ay =ya}Note tha for any a E G, we have Z(G) < CG(a). In other words, the center is contained in the centralizer of any element. Definition 3.4 A group G is called cyclic if? There exists an element a E G such that G=. Any such element is called a generator of G. Theorem 3.9: Let G be a group and a E G. Then for all i,j E Z, we have: 1. If a has infinite order, then a^i=a^j IFF i=j2. If a has finite order |a|=n, then a^i=a^j IFF n divides i-j Theorem 3.14 Let G= be a cyclic group with generator a of order |G|=|a|=n. Then? for any elemn a^s E G we have |a^n|=n/gcd(n,s) Theorem 3.21 Every subgroup of a cyclic group G is? Cyclic Theorem 3.24 Let G= be a cyclic group of order n. Then: 1. the order |H| of any subgroup H of G is a divisor of n=|G|2. For each positive integer d that divides n there exists a unique subgroup of order d, the subgroup H= AuthorJorge732 ID259217 Card SetMath 545 midterm 1 DescriptionMath 545 first Midterm Updated2014-01-30T07:39:30Z Show Answers