Lagrange's Theorem: Clarifying | | Meaning

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The order of a subgroup H of a group G, denoted as |H|, is a factor of the order of G. This means that the order of G is a multiple of the order of H. For example, if |G| = 24 and |H| = 6, then there are 4 different cosets of H in G, meaning that m = 4. This also means that 4|H| = |G|. In the conversation, they are discussing the order of a subgroup in relation to the order of the group, not the number of elements in the group. And yes, you are correct that the symmetric group of 4 (S4) has 4! elements, but that is
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Firepanda
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The definition I have is:

Leg G be a finite group and let H be a subgroup of G. Then the order of H is a factor of the order of G. More precisely, |G|=m|H| where m is the number of different cosets of H in G.

Can someone clarify what the | | means?

I thought it was how many elements are in a group, such as, the symmetric group of 4 (S4) has 4! elements, so |S4| = 24.

I have an example saying the S4 cannot have a subgroup of order 5 since |S4| = 24 which is not an exact multiple of 5.

But 24 is the number of elements in S4, not the order of the group.. So why are they saying the subgroup can't have order 5 because of the number of elements in S4? Surely we should be finding the order of S4 instead to se if there is a subgroup of order 5 in the group.

Thanks
 
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The order of a group G is the number of elements in the group G, which is denoted as |G|. I don't know what is confusing you.
 

Related to Lagrange's Theorem: Clarifying | | Meaning

What is Lagrange's Theorem and what does it clarify?

Lagrange's Theorem, also known as the First Isomorphism Theorem, is a fundamental result in abstract algebra that clarifies the relationship between the structure of a group and its subgroups. It tells us that the number of cosets of a subgroup is equal to the index of the subgroup, and provides a way to construct a new group from a given group by quotienting out a normal subgroup. This helps us understand the underlying structure of a given group and its subgroups.

What is a subgroup and how is it related to Lagrange's Theorem?

A subgroup is a subset of a group that is itself a group with respect to the group operation. Lagrange's Theorem states that the order of a subgroup must divide the order of the original group. In other words, if a subgroup has a certain number of elements, then the original group must have a multiple of that number of elements. This relationship helps us understand the size and structure of subgroups within a larger group.

How is Lagrange's Theorem used in abstract algebra?

Lagrange's Theorem is a fundamental tool in abstract algebra and is used to prove many other important theorems. It is also used to classify groups and understand their subgroups. Additionally, the theorem has applications in other areas of mathematics, such as number theory and geometry.

What is the significance of Lagrange's Theorem in group theory?

Lagrange's Theorem is one of the most important and fundamental results in group theory. It provides a key link between the structure of a group and its subgroups, and helps us understand the underlying properties of groups. Additionally, many other important theorems in group theory, such as the Sylow theorems, are built upon Lagrange's Theorem.

Can Lagrange's Theorem be extended to other mathematical structures?

Yes, Lagrange's Theorem has been extended to other algebraic structures, such as rings and fields. In these cases, it is known as the First Isomorphism Theorem for rings and fields. The concept of index and cosets can also be applied to other structures, such as vector spaces and modules, allowing for a generalization of Lagrange's Theorem to these contexts.

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