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9 Cards in this Set

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  • Back
What defines a group?
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.
what is the order of a group?
The number of elements it has
what is the order of an element a?
the least positive integer n such that a^n = identity
What is a subgroup?
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.)
How is the order of a subgroup related to the order of a group?
it divides it. In other words, if O(g)=m and O(s)=n then m/n is an integer.
How do we classify a group whose operation doesn't commute?
We call it non-abelian
What is a cyclic group?
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.
What is formed when you apply successive composition of the operation defining the group to a non-primitive element of the group?
You get a cyclic subgroup
Can groups with orders that are prime have non-primitive elements?

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.