Problem concerning cyclic groups.

[jmjlt88]

The question states:
"Let G be a group and let G_n={gⁿ|g ε G}. Under what hypothesis about G can we show that G_n is a subgroup of G?The set G_n is taking each element of G and raising it to a fixed number. I started my investigation by examining what happens if I take n=3 and considering the groups Z₄ and the Klein 4-group. I noticed that G_n did not create a subgroup with the integers modulo 4, but it did create one with the Klein 4-group. Thus, I believe that in order for G_n be a subgroup, we need the condition that G must NOT be cyclic.

That being said, I have a proof, which seems to work, but nowhere did I use the fact that G is not cyclic. I have an outline below.

Thrm: If G is a group and is not cyclic, then G_n={gⁿ|g ε G} is a subgroup.

Proof (SKETCH): Let G be a group and suppose G is not cyclic. We wish to show G_n={gⁿ|g ε G} is a subgroup.
*Associativity*
... inherited from G
*Identity*
... The identity element e from G is in G_n, and e raised to some fixed number n is still e.
*Inverse*
... a and a^-1 are in G... aⁿ ε G_n .. a^-1n= a^-n ε G_n ... aⁿa^-n=a^-naⁿ=eThe above is just an outline. But, even with all the details, I never use the fact that G is not cyclic. I must be doing something wrong. Please help! Thanks

 

[jmjlt88]

^ Forgot to show closure. Let a,b ε G_n. Then a=g₁ⁿ and b=g₂ⁿ for some g₁,g₂εG. Then,
ab = g₁ⁿg₂ⁿ=(g₁g₂)ⁿ. Hence ab ε G_n

 

[Dick]

I think there's some confusion here. If you mean Z4 to be the integers mod 4 that's not a group under multiplication. You want the group action to be multiplication. A cyclic group of order 4 would be {e,a,a^2,a^3} with the relation a^4=e.

 

[jmjlt88]

I was completely wrong when I went about investigating Z⁴. Ignore that completely...


I was also completely wrong about that G having to be NONCYCLIC. It can be cyclic. G must be ABELIAN. And, I did end up using this fact ... Right here ..
ab = g₁ⁿg₂ⁿ=(g₁g₂)ⁿ

If G wasn't commutative, then I wouldn't be able to show that G_n is closed.

 

[Dick]

I was completely wrong when I went about investigating Z⁴. Ignore that completely...


I was also completely wrong about that G having to be NONCYCLIC. It can be cyclic. G must be ABELIAN. And, I did end up using this fact ... Right here ..
ab = g₁ⁿg₂ⁿ=(g₁g₂)ⁿ

If G wasn't commutative, then I wouldn't be able to show that G_n is closed.

That sounds better.