Factor Groups: What am I thinking about wrongly here?

[PsychonautQQ]

say K is normal in G hence we have a factor group G/K.

let g be an element of G where |g| = n.

so Kg^n = K since g^n = 1.
and using the properties of factor groups, we know Kg^n = (Kg)^n
hence (Kg)^n = K
So we know that the order of Kg divides n.

Is this correct thinking? Factor groups are trippin me out

 

[Boorglar]

Yes, this looks correct. In fact, this shouldn't be surprising, since by Lagrange's theorem, the order of the factor group G/K is |G|/|K| which divides |G| = n.
Thus the order of any element of G/K must divide |G/K| which divides n. You can easily show that a divisor of a divisor of n is also a divisor of n.

 

[PsychonautQQ]

you said |G| = n. I originally said it was an element of G, |g| = n. Doesn't |G| = |g| iff G = <g> (G is a cyclic group generated by g?) The question doesn't say G is cyclic.

 

[Boorglar]

Oh my bad I misread that the order of G was n. In that case the statement that |Kg| divides n does not directly follow from what I said, but it is still true by what you showed.

 

[blue_raver22]

Your thinking is mostly correct, but there are a few things to consider. First, when you say "K is normal in G," it means that K is a normal subgroup of G, not just a subgroup. This means that for any element k in K and any element g in G, the product kg is also in K. This is important because it allows us to define the factor group G/K, where the elements are the cosets of K in G.

Now, let's look at your statement that Kg^n = K since g^n = 1. This is true, but it's important to note that g^n = 1 only if g has finite order. If g has infinite order, then g^n may not equal 1.

Additionally, your statement that (Kg)^n = K is not necessarily true. In fact, it's only true if K is a normal subgroup of G. If K is not normal, then (Kg)^n may not equal K.

Finally, your statement that the order of Kg divides n is correct, but it's important to note that this is a consequence of the properties of factor groups, not a definition of factor groups.

Overall, your thinking is on the right track, but it's important to be precise with definitions and properties when dealing with abstract concepts like factor groups. Keep exploring and asking questions, and you'll continue to deepen your understanding!