Question about order of an element

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In summary, the conversation discusses the concept of cyclic subgroups in a group and how every element can generate a cyclic subgroup, regardless of its order.
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epr2008
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I think that I just noticed something that I probably should have noticed a lot earlier in my Abstract Algebra class:

If G is a group which contains some g of finite order, then by definition g^k=e for some integer k. Does this mean that each element of finite order in G can be written as the generator of some some cyclic subgroup of G?
 
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Every element of a group generates a cyclic subgroup of the larger group, and the order of the element doesn't need to be finite for the resulting subgroup to be cyclic. Cyclic just means that it's generated by one element.
 

FAQ: Question about order of an element

What is the order of an element?

The order of an element is the smallest positive integer n such that the element raised to the power of n results in the identity element. In other words, it is the number of times the element must be multiplied by itself to equal the identity element.

How is the order of an element related to the order of a group?

The order of an element is always a factor of the order of the group. This means that the order of the group will always be a multiple of the order of any element within that group.

Can an element have more than one order?

No, an element can only have one order in a given group. However, the order of an element may be different in different groups.

How is the order of an element related to cyclic groups?

If an element has an order that is equal to the order of the group, then the group is considered a cyclic group. This means that the element, when multiplied by itself, will generate all other elements in the group.

How can I calculate the order of an element?

The order of an element can be calculated by finding the smallest positive integer n such that the element raised to the power of n results in the identity element. This can be done through a process of trial and error or by using mathematical formulas and algorithms.

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