group theory

Group THeory

BingoYou must write the slide number on the clue to get credit

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