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