Subgroup Order in Groups of Divisible Orders: Proof or Counterexamples?

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In summary, 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. It is a smaller version of the larger group and has a similar algebraic structure. The order of a subgroup refers to the number of elements in the subgroup, and it can be determined by counting the elements or finding the greatest common divisor. A subgroup can have a different order than the larger group, with a common case being a smaller order due to it being a subset of the larger group.
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arshavin
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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?
 
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No - one reason to see immediately why, is because if it was true, then Lagrange's theorem should be a two way implication.

Simplest example via wikipedia is [tex]\mathbb{A}_4[/tex] with has order 12 and no subgroup of order 6.
 
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FAQ: Subgroup Order in Groups of Divisible Orders: Proof or Counterexamples?

What is a subgroup?

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.

How is a subgroup related to a 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.

What is the order of a subgroup?

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.

How is the order of a subgroup determined?

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.

Can a subgroup have a different order than the larger group?

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.

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