Whenever a question asks you to prove a group is abelian, remember this method:

1. Let A, B be arbitrary elements of the group 2. Calculate AB 3. Calculate BA 4. show they are equal, if they are not then its not abelian

A set with an associative operation, identity and inverses

A non empty subset H of G that is itself a group under the same operation

What is the subgroup test?

H is a subgroup if H is non empty and ab^-1 is an element of H for all a, b is an element of H

What is a normal subgroup

H is a sub group of G if gHg^-1 = H for every g is an element of G

What is an abelian group

A group where ab = ba for all elements a, b

What is a cyclic group

A group generated by one element G = = {a^n: n is an element of Z}

State Lagranges theorem

For finite G and H is less than or equal to G the order of H divides G

What is a group homomorphism

A map from G -> H preserving operations (ab) = (a)(b)

What is group isomorphism

A bijective homomorphism it means the groups have the same structure

state Cayley's theorem

Every group isomorphic to a subgroup of a permutation group

Check closure, associativity, identity, and inverses

Prove it is a subgroup

H is non empty and ab^-1 is a element of H for every a, b is an element of H

Take arbitrary a, b in G and show ab = ba

Prove it is NOT abelian

Find just one pair where ab not equal to ba

Find one element that generates every element of the group.

Prove it is NOT cyclic

Show that no element can generate the entire group. For a finite group, you can check the elements and show each generates only a proper subgroup.

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Abstract Algebra

Question	Answer
Whenever a question asks you to prove a group is abelian, remember this method:	1. Let A, B be arbitrary elements of the group 2. Calculate AB 3. Calculate BA 4. show they are equal, if they are not then its not abelian
What is a group?	A set with an associative operation, identity and inverses
What is a subgroup?	A non empty subset H of G that is itself a group under the same operation
What is the subgroup test?	H is a subgroup if H is non empty and ab^-1 is an element of H for all a, b is an element of H
What is a normal subgroup	H is a sub group of G if gHg^-1 = H for every g is an element of G
What is an abelian group	A group where ab = ba for all elements a, b
What is a cyclic group	A group generated by one element G = <a> = {a^n: n is an element of Z}
State Lagranges theorem	For finite G and H is less than or equal to G the order of H divides G
What is a group homomorphism	A map from G -> H preserving operations (ab) = (a)(b)
What is group isomorphism	A bijective homomorphism it means the groups have the same structure
state Cayley's theorem	Every group isomorphic to a subgroup of a permutation group
Prove it is a group	Check closure, associativity, identity, and inverses
Prove it is a subgroup	H is non empty and ab^-1 is a element of H for every a, b is an element of H
Prove it is abelian	Take arbitrary a, b in G and show ab = ba
Prove it is NOT abelian	Find just one pair where ab not equal to ba
Prove it is cyclic	Find one element that generates every element of the group.
Prove it is NOT cyclic	Show that no element can generate the entire group. For a finite group, you can check the elements and show each generates only a proper subgroup.

Created by: jayne_

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