Abstract Algebra (Gallian) Exam Questions and Answers 100% Pass

Abstract Algebra (Gallian) Exam Questions and Answers 100% Pass divisor -Answer-t is a divisor if there is an integer u such that s = tu multiple -Answer-s is a multiple if there is an integer u such that s = tu division algorithm -Answer-let a and b be integers with b > 0, then there exist unique integers q and r with the property that a = bq + r, where 0 ≤ r < b order of a group |G| -Answer-e number of elements of a group is called its order order of an element |g| -Answer-the order of an element g in group G is the smallest positive integer n such that gⁿ = e (addition it is ng = 0). if no such integer exists, we say that g has infinite order subgroup -Answer-if a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G.. notation H ≤ G 1 step subgroup test [theorem] -Answer-let G be a group and H a nonempty subset of G. Then, H is a subgroup of G if a(b^-1) ∈ H whenever a, b ∈ H. 2 step subgroup test [theorem] -Answer-let G be a group and H a nonempty set of G. Then, H is a subgroup of G if ab ∈ H whenever a, b ∈ H and and a^(-1) ∈ H. finite subgroup test [theorem] -Answer-let H be a nonempty finite subset of G. Then, H is a subgroup of G if H is closed under the operation of G. <a> -Answer-= {a^n | n ∈ Z} <a> is a subgroup [theorem] -Answer-let G be a group and let a ∈ G. Then, <a> is a subgroup of G. center is a subgroup [theorem] -Answer-the center of a group G is a subgroup of G. center of a group -Answer-center, Z(G) of a group G is the subset of elements that commute with every element in G. Z(G) = {a ∈ G | ax = xa for all x in G} centralizer of a in G -Answer-let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. C(a) = {g ∈ G | ga = ag} cyclic -Answer-a group G is a called cyclic if there is an element a in G such that G = {a^n | n ∈ Z}. G = <a> generator -Answer-a group G is a called cyclic if there is an element a in G such that G = {a^n | n ∈ Z}; such an element a is called a generator of G criterion for a^i = a^j [theorem] -Answer-let G be a group, and let a belong to G. If a has infinite order, then all distinct powers of a are distinct group elements. If a has finite order, say, n, then <a> = {e, a, a^2, ... , a^(n-1)} and a^i = a^j if and only if n divides i - j. |a| = |<a>| [corollary] -Answer-for any group element a, |a| = |<a>|. a^k = e implies that |a| divides k [corollary] -Answer-Let G be a group and let a be an element of order n in G. If a^k = e, then n divides