# Math 545 midterm 1

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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 G
• 2. Associativity: For any  a, b, c E G, we have a*(b*c)=(a*b)*c
• 3. 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.
2. Proposition 1.18
Basic group properties: For any group G:
• 1. The identity element of G is unique
• 2. For each a E G, the inverse a^-1 is unique
• 3. For any a E G, (a^-1)^-1=a
• 4. For any a, b E G, (ab)^-1=b^-1a^-1
• 5. 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
3. 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 H
•       For every a, b E H, a-b E H
4. 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 H
• 2. inverse: For every b E H, b^-1 E H
5. Definition 2.13
Let G be a group and a E G. Then we define
• <a>={a^n|n E Z}
• If the group operation is written as addition,
• <a>={na| n E Z}
6. 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
7. 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.
8. Theorem 2.24:
The Center Z(G) of a group is?
a subgroup of G.
9. 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.
10. Definition 3.4
A group G is called cyclic if?
• There exists an element a E G such that
• G=<a>. Any such element is called a generator of G.
11. 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=j
• 2. If a has finite order |a|=n, then a^i=a^j IFF n divides i-j
12. Theorem 3.14
Let G=<a> 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)
13. Theorem 3.21
Every subgroup of a cyclic group G is?
Cyclic
14. Theorem 3.24
Let G=<a> 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=<a^n/d>

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 Author: Jorge732 ID: 259217 Filename: Math 545 midterm 1 Updated: 2014-01-30 07:39:30 Tags: Math 545 Folders: Description: Math 545 first Midterm Show Answers:

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