Group THeory
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Transcript of Group THeory
Group THeory
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Rules and Rewards
• The following slides have clues• Each clue may refer to a theorem or term on your
bingo card• If you believe it does, write the slide number in
the corresponding box• The first student to get Bingo wins 100 points for
their house• Any student to submit a correct card will earn 5
points extra on their test
If is a group, a , then | | [ : ] | |nd G H G G G H H
La Grange’s Theorem
Name the theorem below.
Below is the definition of:
A noncyclic group of order 4
Klein 4 Group
Let be a group and .
mi
1 }n{ | nG
G
n
g
g
G
The definition of this term is below
The order of g
The definition of the term is below
:f G G G
Binary Operation
The permutation below is the _____________ of (1234)
(1432)
inverse
The definition below is called a ______________ ________
1 2 1 2( ) ( ) ( )ff g f g gg
Group Homomorphism
{1,4}
It is the ________________ of {0,3} in 6
Coset
The subgroup below has __________ 5 in D5
{(25)(34), }e
Index
1 1( )Hf
If f is a group homomorphism from G to H, then it is the definition of ______________________
Kernel
It is the group of multiplicative elements in Z8
*8
It is an odd permutation of order 4
(1234)
It has 120 elements of order 5
S6
Has a cyclic group of order 8.
It has a trivial kernel
Isomorphism
It is used to show that the order of an element divides the order of the group in which it resides.
The Division Algorithm
The set of all polynomials whose coefficients in the integers, with the operations addition and multiplication, is an example of this.
A ring
It is a set with a binary operation which satisfies three properties.
A group
This element has order 12
(123)(4567)
If f(x) = 3x-1, then the set below is the ________ of 1.
| ( ){ 1}X f xx
Preimage
It is the definition below where R and S are rings.
1 2 1 2
1 2 1 2
:)
such that ( ) ( (
) ( ) ( ))
(
Sf r f rf
f Rr f rr f r fr r
Ring Homomorphism
The kernel of a group homomorphism from G to H is ____________ in G
A normal subgroup
The number 0 in the integers is an example of this
Identity
This element generates a group of order 5
(12543)
It is a way of computing the gcd of two numbers
The Euclidean Algorithm
A function whose image is the codomain
Surjective
It is a commutative group
Abelian
It is a group of order n
Zn
It is a subset which is also group under the same operation
Subgroup
If f: X Y, then it is f(X).
Image
It is the order of 1 in Zmod7.
Seven