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# Category Archives: Group

## The G/Z Theorem

Theorem: Let be a group and let be the center of . If is cyclic, then is Abelian. Proof: Suppose that is cyclic. Then there is some such that . Let . We wish to show that . It follows that there … Continue reading

## Gallian- Ch. 8 #11

Problem Statement: How many elements of order does have? Explain why has the same number of elements of order as . Generalize to the case of . Solution: First let’s find how many elements there are of order in . An element has … Continue reading

## Gallian: Ch. 3 #35

Problem Statement: Prove that a group of even order must have an element of order . Proof: Let be a group such that and consider the set . is the set of all elements of that are not of order . … Continue reading

Posted in Algebra, Group, Math, Order
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