group theory bingo you must write the slide number on the clue to get credit
TRANSCRIPT
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Group THeory
BingoYou must write the slide number on the clue to get credit
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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
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If is a group, a , then | | [ : ] | |nd G H G G G H H
La Grange’s Theorem
Name the theorem below.
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Below is the definition of:
A noncyclic group of order 4
Klein 4 Group
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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
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The definition of the term is below
:f G G G
Binary Operation
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The permutation below is the _____________ of (1234)
(1432)
inverse
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The definition below is called a ______________ ________
1 2 1 2( ) ( ) ( )ff g f g gg
Group Homomorphism
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{1,4}
It is the ________________ of {0,3} in 6
Coset
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The subgroup below has __________ 5 in D5
{(25)(34), }e
Index
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1 1( )Hf
If f is a group homomorphism from G to H, then it is the definition of ______________________
Kernel
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It is the group of multiplicative elements in Z8
*8
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It is an odd permutation of order 4
(1234)
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It has 120 elements of order 5
S6
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Has a cyclic group of order 8.
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It has a trivial kernel
Isomorphism
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It is used to show that the order of an element divides the order of the group in which it resides.
The Division Algorithm
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The set of all polynomials whose coefficients in the integers, with the operations addition and multiplication, is an example of this.
A ring
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It is a set with a binary operation which satisfies three properties.
A group
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This element has order 12
(123)(4567)
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If f(x) = 3x-1, then the set below is the ________ of 1.
| ( ){ 1}X f xx
Preimage
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It is the definition below where R and S are rings.
1 2 1 2
1 2 1 2
:
)
such that
( ) ( (
) ( ) ( )
)
(
S
f r f r
f
f R
r f r
r f r fr r
Ring Homomorphism
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The kernel of a group homomorphism from G to H is ____________ in G
A normal subgroup
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The number 0 in the integers is an example of this
Identity
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This element generates a group of order 5
(12543)
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It is a way of computing the gcd of two numbers
The Euclidean Algorithm
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A function whose image is the codomain
Surjective
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It is a commutative group
Abelian
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It is a group of order n
Zn
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It is a subset which is also group under the same operation
Subgroup
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If f: X Y, then it is f(X).
Image
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It is the order of 1 in Zmod7.
Seven