- #1
arshavin
- 21
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If G is a group of order n, and n is divisible by k. Then must G have a subgroup of order k?
proof or counterexamples?
proof or counterexamples?
A subgroup is a subset of a larger group that follows the same algebraic rules and contains all of the identity and inverse elements of the larger group.
A subgroup is a smaller version of a larger group, with similar properties and operations. It is a part of the larger group and follows the same algebraic structure.
The order of a subgroup refers to the number of elements in the subgroup. It can be thought of as the size or cardinality of the subgroup.
The order of a subgroup is determined by counting the number of elements in the subgroup. It can also be determined by finding the greatest common divisor of all the elements in the subgroup.
Yes, a subgroup can have a different order than the larger group. In fact, it is common for a subgroup to have a smaller order than the larger group. This is because a subgroup is a subset of the larger group and may not contain all of the elements of the larger group.