Abstract Algebra THEOREMS

Let R be an equivalence relation defined on a set S and define [a]= {b in S| bRa}. Then _______________

True or False, if S is a set closed under a binary operation * , then S contains more than one identity

Let S be a set closed under a binary operation *, with identity e, where * is ([blank_start]Answer 1[blank_end]). Then x (in S) has ([blank_start]Answer 2[blank_end]) Inverses.

Let S be a set, closed under an associative binary operation *, with identity e. Then, if x^-1 and y^-1 exist then

Suppose n is a Irrational Number and S a set closed under associative binary operation * then x^n = x * x * x * ... * x (n terms)

A group is a pair (S,*) where S is a non-empty set and * a binary operation on S such that; S is [blank_start]ANSWER 1[blank_end] under *, * is [blank_start]ANSWER 2[blank_end] on S, S has [blank_start]ANSWER 3[blank_end] w.r.t. to * denoted e, every element in S has an [blank_start]ANSWER 4[blank_end], denoted x^-1 in S

Tick the properties required for the General Linear Group of degree n

What are the properties of an ISOMETRY

If P is the set of points representing a figure in space, and G is the set of all isometries that map P onto itself, then (G, *) is a group called the Symmetry Group Of P

What does the following mean in words?

Let (G, *) be a group with identity element e and consider some element a in G. What does it mean to say that a has period w?

Complete the following definition: Let (G,*) be a group and H be a [blank_start]subset[blank_end] of G. Then, H is a [blank_start]subgroup[blank_end] of G [blank_start]if and only if[blank_end] (H,*) is a group.

True or False: If (H,*) is a subgroup of (G,*) then; i) they have the same identity ii) the inverse of x in H is the inverse of x in G

Complete the definition: If H is a [blank_start]non-empty[blank_end], finite, subset of the [blank_start]elements[blank_end] of a group (G,*), then (H,*) is a subgroup of (G,*) [blank_start]if and only if[blank_end] H is [blank_start]closed[blank_end] under *.

Let (G,*) be a group and x be in G. Then _____________ forms a subgroup of (G,*) called the cyclic subgroup generated by x, denoted <x>

Select the properties for which a binary operation is uniquely defined

True or False, if (G,*) is a cyclic group with generator a, denoted <a>. Then |G| = period of a

Which ONE of the following is used to describe a LEFT COSET?

Which of the following describes that a) the period of the coset is the period of the set b) each element can only be in one coset ( G is a finite group and H a subgroup). SELECT TWO

[blank_start]LAGRANGE[blank_end]'S THEOREM: Let (H,*) be a subgroup of a [blank_start]finite[blank_end] group (G,*). Then the number of [blank_start]elements[blank_end] in H divides the number of elements in G; that is IHI | IGI

Let (G,*) be a finite group of order n and let g be an element with period k. Then i) k|n (ie. the period of an element divides the order or the group) and ii) g^n = e

Complete the theorem: Let (G,*) be a [blank_start]finite[blank_end] group of order p, where p is [blank_start]prime[blank_end]. Then, (G,*) is [blank_start]cyclic[blank_end]

Which of the following are properties of a homomorphism?

Which of the following hold for an Isomorphism of groups f: G1 -> G2