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Gear300
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Since certain operations are not commutative, when a group G = <a, b>, does the order matter (so that <a, b> is not necessarily equal to <b, a>)?
A cyclic subgroup is a subgroup of a group that is generated by a single element. This means that all elements in the subgroup can be obtained by repeatedly applying the group operation to the generator element.
A regular subgroup can be generated by multiple elements, while a cyclic subgroup can only be generated by a single element. Additionally, all cyclic subgroups are abelian (commutative), while regular subgroups may not be.
Yes, a cyclic subgroup can be infinite. This happens when the generator element has infinite order, meaning that it can be multiplied with itself an infinite number of times without ever returning to the identity element.
A cyclic group is a group in which every element can be generated by a single element. In other words, every element in the group belongs to a cyclic subgroup.
Cyclic subgroups can be useful in mathematical proofs because they often have well-defined properties that can be used to simplify calculations or prove certain statements. For example, if a group is known to be cyclic, then its order (number of elements) can be easily determined.