Can Order and Size be Equivalent?

  • Thread starter Homo Novus
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In summary: The order of a factor group is the smallest positive integer n such that the group is a subgroup of the factor group generated by the given element.
  • #1
Homo Novus
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Is "order" = "size"?

I'm really confused... we refer to the "size" of a group as the "order"... But are they really equivalent? Can we prove that simply?

Example:
Say we have a group G and a subgroup H. Then take one of the cosets of H, say Hx... It has order = o(G)/o(G/H). But does this mean that in G/H, (Hx)^[o(G)/o(G/H)] = e = H?? Or do elements have different orders in different groups? I'm so confused.

Oh, and this whole factor group thing... Does this only apply to normal subgroups?
 
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  • #2


You seem to be talking about two different definitions of "order". The "order of a group" is the number of elements in the group- its "size". The "order of an element" of group G is the order of the subgroup of G generated by the element. In particular if the order of an element, x, is n then x^n= e, the identity of G. I don't know what you mean by a subgroup, (Hx), to a power.
 
  • #3


Homo Novus said:
I'm really confused... we refer to the "size" of a group as the "order"... But are they really equivalent? Can we prove that simply?

No, they are not equivalent. One definition of order means the "size" of the group. The other definition of order is the order of an element g: it is the smallest positive integer n such that [itex]g^n=e[/itex].

Example:
Say we have a group G and a subgroup H. Then take one of the cosets of H, say Hx... It has order = o(G)/o(G/H).

Why does it have that order? Sure, the coset has |H| elements. But the order is the smallest positive number n such that [itex](Hx)^n=H[/itex]. This is in general not o(G)/o(G/H).


Oh, and this whole factor group thing... Does this only apply to normal subgroups?

Yes.
 

1. Can order and size be equivalent in all situations?

No, order and size can be equivalent in some situations, but not all. It depends on the context and variables involved.

2. How do we determine if order and size are equivalent?

To determine if order and size are equivalent, we need to compare the objects or elements in question and consider factors such as their relative position, quantity, and physical characteristics.

3. Is size always a reflection of order?

No, size is not always a reflection of order. While size can sometimes be a factor in determining order, there are other factors such as arrangement and hierarchy that also play a role.

4. Can order and size change over time?

Yes, order and size can change over time. As objects or elements are added or removed, their relative position and quantity can change, affecting both order and size.

5. How does perception affect our understanding of order and size?

Perception can greatly influence our understanding of order and size. Different perspectives and biases can impact how we interpret the relationship between order and size in a given situation.

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