Prove Lagrange’s Theorem for left cosets

[AutGuy98]

Hey guys,

Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in light of that, I actually have some more questions for an exercise set that I have to do for one of my classes and I'm really unsure how to begin doing them. There are four of them and they all require proofs to some degree. Anyway, I was going to make one post and put all four parts of the same question in it (i.e. 2(a),2(b),2(c), and 2(d)), but was unsure whether or not it would be allowed here. So, for those reasons and to play it safe rather than try to do so, here is the first part that I've been having trouble with. Any help here is, once again, greatly appreciated and will leave me forever further in your gratitude.

Question: 2(a): "Prove Lagrange’s Theorem for left cosets."

Again, I have no idea where to start with this, so any help is extremely gracious and appreciated.

P.S. If possible at all, I'd need help on these by tomorrow at 12:30 E.S.T., so please try to look this over at your earliest conveniences. Thank you all again for your help with everything already.

 

[Euge]

Hello again, AutGuy98! (Wave)

How is Lagrange's theorem stated in your text?

 

[AutGuy98]

Hello again, AutGuy98! (Wave)

How is Lagrange's theorem stated in your text?

In the book, it says, "You may have noticed that the order of a subgroup H of a finite group G seems always to be a divisor of the order of G. This is the theorem of Lagrange." Please let me know if this helps or not. Also, thank you for your help on 2(b). It is very much appreciated!

 

[Euge]

Define an equivalence relation on $G$ by declaring $x\sim y$ for $x,y\in G$ iff there exists an $h\in H$ such that $y = xh$. The equivalence class of an element $x\in G$ is the left coset $xH$, and it follows that the left cosets of $H$ partition $G$. Given a left coset $xH$ of $H$ in $G$, there is a one-to-one correspondence $\Delta : H \to xH$ given by $\Delta(h) = xh$. Therefore, $xH$ has $\lvert H\rvert$ elements. Since each coset of $H$ has $|H|$ elements and the cosets partition $G$, then $|G| = (G : H)|H|$, showing that $|H|$ divides $|G|$.