Why is the smallest subgroup of G containing A and B equal to G itself?

In summary, a group is a set of elements that follow a specific set of rules and operations, and a subgroup is a smaller set within a group that also follows the same rules and operations. An example of a group is the set of integers under addition, and an example of a subgroup within this group is the set of even integers. Groups and subgroups are related in that a subgroup is a subset of a larger group that follows the same rules and operations. A subgroup can also be a group if it satisfies all the requirements of a group. Real-life examples of groups and subgroups include elements and their isotopes in chemistry, the set of rational numbers in mathematics, and communities in social science.
  • #1
blahblah8724
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In an example it says that, if [itex] |G| = 15 [/itex] and [itex] G [/itex] has subgroups [itex] A,B [/itex] of [itex] G [/itex] with [itex] |A| = 5 [/itex] and [itex] |B| = 3 [/itex], then [itex] A \cap B [/itex] must equal [itex] \{e_G\} [/itex] and the smallest subgroup of [itex] G [/itex] containing both [itex] A [/itex] and [itex] B [/itex] is [itex] G [/itex] itself. Could anyone explain why? Thanks!
 
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  • #2
Hint: Lagrange
 

FAQ: Why is the smallest subgroup of G containing A and B equal to G itself?

What is a group and subgroup?

A group is a set of elements that follow a specific set of rules and operations, such as addition or multiplication. A subgroup is a smaller set within a group that also follows the same rules and operations.

What is an example of a group and subgroup?

An example of a group is the set of integers under addition, where 1, 2, 3, etc. can be added together to get another integer. An example of a subgroup within this group is the set of even integers, where 2, 4, 6, etc. can be added together to get another even integer.

How are groups and subgroups related?

A subgroup is a subset of a larger group that follows the same rules and operations. This means that any operation performed on elements within the subgroup will still result in an element within the subgroup, making it a smaller version of the larger group.

Can a subgroup also be a group?

Yes, a subgroup can also be a group if it satisfies all the requirements of a group. For example, the set of positive integers under addition is a subgroup of the group of all integers under addition, but it is also a group on its own.

What are some real-life examples of groups and subgroups?

In chemistry, elements and their isotopes can be grouped into subgroups based on their atomic properties. In mathematics, the set of rational numbers is a subgroup of the set of real numbers. In social science, communities can be considered as subgroups within a larger society or culture.

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