# Is order = size ?

1. Oct 31, 2012

### 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?

2. Oct 31, 2012

### HallsofIvy

Staff Emeritus
Re: Is "order" = "size"?

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. Oct 31, 2012

### micromass

Staff Emeritus
Re: Is "order" = "size"?

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 $g^n=e$.

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).

Yes.