# Abstract Algebra Key Definitions Theorems

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1. State the Well Ordering Principle
Every nonempty set of positive integers contains a smallest element.
2. State the Division Algorithm
Let a and b be integers where b > 0. Then there exists unique integers q&r such that: a = bq + r where 0< r < b
3. Linear Combination
For any nonzero integers a and b, there exists integers s & t such that gcd(a,b) = as + bt. Moreover, gcd(a,b) is the smallest positive integer of the form as + bt.
4. If a and b are relatively prime...
... then there exists integers s and t such that as + bt = 1.
5. Define Euclid's Lemma
If p is a prime that divides ab, then p divides a or p divides b.
6. Define the GCD
• Greatest Common Divisor
• The GCD of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer gcd(a,b).
7. Define the LCM
• Least Common Multiple
• The LCM of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. We denote this as lcm(a,b).
8. Define an Equivalence Relation
• An equivalence relation on a set S is a set of ordered pairs R of S such that
• ~(a,a) is an element of R for all a element of S (reflexive)
• ~(a,b) is an element of R then (b,a) is an element of R (symmetric)
• ~If (a,b) is an element of R and (b,c) is an element of R then (a,c) is an element of R (Transitive)
9. Partition of a Set
A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.
10. Define a function
A function (mapping) is a rule that assigns each a of A a unique element b of B.
11. Dihedral Group
a gropu denoted Dn of order 2n
12. What is a group?
Let G be a set under a binary operation. A group assigns to each ordered pair (a,b) of elements of G an element in G denoted by ab.
13. What is a binary operation?
A binary operation on set G is a function that assigns each ordered pair of elements of G an element of G.
14. What three properties must a group satisfy?
• 1. Associativity. i.e. (ab)c = a(bc) for all a,b,c in G.
• 2. Identity. i.e. there is an element called e such that aeea = a for all a in G.
• 3. Inverses. i.e. there is an element b such that ab=ba=e.
15. What does it mean to say a group is Abelian?
The group has the property that ab = ba for every pair of elements a and b.
16. What is the Uniqueness of the Identity theorem?
In a group G, there is only one identity element.
17. State the Cancellation Theorem
Right and Left cancellation holds such that ba=ca implies b=c, and ab=ac implies b=c.
18. State the Uniqueness of Inverses Theorem
For each element a in a group G there is a unique element b in G such that ab=ba=e.
19. What is the Socks-Shoes-Property?
• For group elements a and b,
• (ab)-1 =b-1a-1
20. Define the Order of a Group
The order of a group (finite or infinite) is the number of elements in this group. We denote the order of group G as |G|.
21. Define the Order of an Element
Let G be a group and g be an element of G. The order of an element g in G is the smallest possible integer n such that gn=e. If no such n exists then we say g has infinite order. Denoted by |g|.
22. Define a Subgroup
If a subset H of a group G is also a group under the same operation of G, we say that H is a subgroup of G. Denoted H<G
23. What are the steps in the One-Step Subgroup test?
• 1. Identify the property P that distinguishes the elements of H
• 2. Prove that the identity has the property P
• 3. Assume that two elements a and b have property P
• 4. Use the assumption that a and b have property P to show that ab-1 also has property P
24. State the One-Step Subgroup test
• Let G be a group and H a nonempty subset of G.
• (multiplication) If ab-1 is in H whenever a and b are in H, then H is a subgroup of G.
• (additive) If a - b is in H whenever a and b are in H, then H is a subgroup of G.
25. State the Two-Step Subgroup Test
• Let G be a group and let H be a nonempty subset of G.
• If ab is in H whenever a and b are in H, and a-1 is in H whenever a is in H, then H is a subgroup of G.
26. State the Finite-Subgroup Test

When is this used?
Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G.

used when we have finite groups
27. What is <a> ?
• Cyclic Group - Let G be a group, and let a be any element of g. Then, <a> is a subgroup denoted by
• <a> = {an | n is an element of Z}
28. How do we denote the center of a group G?
Z(G)
29. Define the Center of a Group
• The center of a group G is the subset of elements in G that commute with every element of G.
• i.e. Z(G) = {a in G | axxa for all x in G}

The center of a group G is a subgroup of G.
30. Define the Centralizer
• Let a be a fixed element of a group G. The centralizer of a in G is the set of all elements in G that commute with a. Denoted C(a).
• i.e. C(a) = {g in G | ga = ag}

The centralizer is a subgroup
31. What does it mean to say a group is Cyclic?
• A group is cyclic such that there exists an element a in G such that G = {a| n is an element of Z}
• denoted <a>
32. What is a Generator?
An element a is a generator of G when an = G
 Author: popolisj ID: 285228 Card Set: Abstract Algebra Key Definitions Theorems Updated: 2014-10-09 18:36:53 Tags: Abstract Algebra Theorems Folders: Description: Review for a Abstract Algebra Test Show Answers: