Can Order and Size be Equivalent?

[Homo Novus]

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?

 

[HallsofIvy]

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.

 

[micromass]

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 $g^n=e$.

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 $(Hx)^n=H$. 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.