I have:

is a surjective homomorphism of groups.

How do I verify this statement:

If and commute, then so do and

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- Oct 22nd 2010, 07:07 AMMichaelMathhomomorphism / commutative property
I have:

is a surjective homomorphism of groups.

How do I verify this statement:

If and commute, then so do and - Oct 22nd 2010, 07:13 AMAckbeet
What does the notation mean? Is that just the homomorphism applied to thus:

- Oct 22nd 2010, 07:17 AMMichaelMath
yep

- Oct 22nd 2010, 07:19 AMAckbeet
Ok, so

- Oct 22nd 2010, 07:46 AMMichaelMath

That simple? (sorry I am horrible at this stuff) - Oct 22nd 2010, 07:50 AMAckbeet
As far as I know, that's correct. I'm not entirely sure where you use the surjective (onto) property here. You use the homomorphism property twice. You use the fact that the homomorphism is defined for all members of X, and you use the closure property of groups. And, of course, the commutativity of the two elements from X.

Incidentally, your \bar{x} looks better than my \overline{x}. I'll have to remember that. - Oct 22nd 2010, 08:29 AMMichaelMath
Hey, feel free to take all of

*my*Latex you want.

I think the surjective part has do with the other 3 questions from the group my question was from. My problem is the textbook I'm using uses different notation than the auld teacher.

I also have

Question: verify is a subgroup of

I figure this is the same as "verify is a subgroup of "

erm... - Oct 22nd 2010, 09:39 AMAckbeet
Aye, laddie. Be you a Scot?

I think you figure correctly. I'm assuming you mean that is a subgroup of . Otherwise, I don't think you'd have a prayer of getting to be a subgroup of .

Well, I'd just go through all 4 group axioms, and verify that they hold for Here, I'm pretty sure you're going to need that is surjective. - Oct 22nd 2010, 09:47 AMMichaelMath
I assume the answer is:

is a subgroup of , so every element, , in , is also an element of .

So every is an element of , and the set of all is a subgroup of

If this is correct, can somebody show me how it should look in proper Mathinese?

p.s. am Irish - Oct 22nd 2010, 09:56 AMAckbeet
You're looking fine there. Being a subgroup gives you structure than just being a subset. So don't be afraid to take advantage of that.

You'd have to prove the following:

1.

2. Closure.

3. Associativity. You might get this one for free, since

4. Identity exists.

5. Inverses exist.

Then you'd be done. So what do you think? - Oct 22nd 2010, 10:14 AMMichaelMath
So i just change my last line from before to: "So every is an element of , and the set of all is a

**subset**of ," and then prove those five properties to show that it is a subgroup? - Oct 22nd 2010, 10:20 AMAckbeet
Right.

- Oct 22nd 2010, 11:48 AMMichaelMath
sorry, still a bit stuck. Can I say stuff like:

Since is a group, under closure:

So,

- Oct 22nd 2010, 11:56 AMAckbeet
Looks good to me!

- Oct 22nd 2010, 02:01 PMMichaelMath