Show that G is abelian, if and only if (gh)^-1 = g^(-1) h^(-1)

[tonit]

Homework Statement

Show that a group G is abelian, if and only if (gh)^{-1} = g^{-1} h^{-1} for all g,h\in G

Homework Equations

The Attempt at a Solution

gh = hg \\⇔ (gh)(gh)^{-1} = hg(gh)^{-1} \\⇔ <br /> <br /> 1 = hg(gh)^{-1} \\⇔h^{-1} = (h^{-1}h)g(gh)^{-1} \\ ⇔g^{-1}h^{-1} = (g^{-1}g)(gh)^{-1}\\ ⇔ (gh)^{-1} = g^{-1}h^{-1}<br /> <br />

The converse is clear from the above by taking the steps backwards

Thus the statement is proved. Correct?

 

[algebrat]

looks kinda messy. your grader is like your boss, you should present your work in an easy to read format.

 

[tonit]

where exactly do you see the mess? So that I can make it clearer?

 

[algebrat]

if gh=hg, take the inverse of both sides, one step, done.

on the other hand, if (gh)^-1=g^-1h^-1, take the inverse of both sides, one step done.

 

[algebrat]

Homework Statement

Show that a group G is abelian, if and only if (gh)^{-1} = g^{-1} h^{-1} for all g,h\in G

Homework Equations

The Attempt at a Solution

gh = hg \\⇔ (gh)(gh)^{-1} = hg(gh)^{-1} \\⇔ <br /> <br /> 1 = hg(gh)^{-1} \\⇔h^{-1} = (h^{-1}h)g(gh)^{-1} \\ ⇔g^{-1}h^{-1} = (g^{-1}g)(gh)^{-1}\\ ⇔ (gh)^{-1} = g^{-1}h^{-1}<br /> <br />

The converse is clear from the above by taking the steps backwards

Thus the statement is proved. Correct?

clear? hmm. concise? no. as every good grader should tell you, this was harder to read than it should have been. it's like in an english class, you don't turn in your first draft. by the end of your first draft, you barely know what you're saying. the nice thing about math, you're not turning in 20 pages. (but sometimes your scratch work may feel like it.)

 

[algebrat]

I'm assuming you know the rule (gh)^-1=h^-1g^-1

 

[tonit]

If gh = hg\\(gh)^{-1} = (hg)^{-1}\Rightarrow (gh)^{-1} = g^{-1}h^{-1}

conversely, if (gh)^{-1} = g^{-1}h^{-1}\\((gh)^{-1})^{-1} = (g^{-1}h^{-1})^{-1}\\\Rightarrow gh = (h^{-1})^{-1}(g^{-1})^{-1} = hg

 

[algebrat]

Yes, that looks a lot cleaner. Now I'm not sure if it was cleaner to prove both directions at the same time, play around with it, but it's right, I'll let you figure out how to pick a final format.

(The way you have it now you don't need the double arrow)

 

[tonit]

yeah. Thanks for the help, I'm learning a lot at this forum really :D