Group Theory Definitions

Front 1

What defines a group?

Back 1

A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:

* Closure : For all a, b in G, the result of a * b is also in G.
* Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
* Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.
* Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element.

Front 2

what is the order of a group?

Back 2

The number of elements it has

Front 3

what is the order of an element a?

Back 3

the least positive integer n such that a^n = identity

Front 4

What is a subgroup?

Back 4

It is a group within a group. It shares the operator of the mother group and some of its elements.

(So the operation on the subgroup can't get you outside of the subgroup.)

Front 5

How is the order of a subgroup related to the order of a group?

Back 5

it divides it. In other words, if O(g)=m and O(s)=n then m/n is an integer.

Front 6

How do we classify a group whose operation doesn't commute?

Back 6

We call it non-abelian

Front 7

What is a cyclic group?

Back 7

It's a group whose elements can all be generated by compositions of operations on the same primitive element.

Hours of the clock under addition is an example.

Front 8

What is formed when you apply successive composition of the operation defining the group to a non-primitive element of the group?

Back 8

You get a cyclic subgroup

Front 9

Can groups with orders that are prime have non-primitive elements?

Back 9

No.

If you could you would get a cyclic subgroup.

The order of the subgroup would divide the order of the group.

This can't happen because the group is prime.