Is H a Subgroup of G in an Abelian Group?

[jr16]

Homework Statement

Let <G, *> be an Abelian group with the identity element, e. Let H = {g ε G| g² = e}. That is, H is the set of all members of G whose squares are the identity.

(i) Prove that H is a subgroup of G.
(ii) Was being Abelian a necessary condition?

Homework Equations

For any subset H of G there are four conditions it must satisfy in order to be classified as a subgroup of G:
1. H is closed under *
2. * is associative
3. The identity, e, of G is also in H
4. For each a in H, the inverse of a, a^-1, is also in H

The Attempt at a Solution

(i) In an attempt to solve this problem, I tried to prove each of the above four conditions:
1. g²*g²
g^{2 + 2}
g⁴
(g²)²
Therefore, H is closed under *

2. (g*g)*g = g*(g*g)
e * g = g * e
g = g
Thus, * is associative

3. e*e = e
e = e
Yes, e is in H

4. g*g = e
(g^-1)*g*g = (g^-1)*e
e * g = (g^-1)
g = (g^-1)
Therefore, (g^-1) is in H

Thus, H is a subgroup of G.
(ii) No, G being Abelian was not a necessary condition (?)


I would really appreciate it if someone could look over my above work. I am not confident in the answer I provided, especially since I did not use the Abelian condition. Thank you for any guidance!

 

[A. Bahat]

1. g2*g2
g2 + 2
g4
(g2)2
Therefore, H is closed under *

Careful here: you are confusing H={g ε G: g^2=e} with a different set, {g^2: g ε G}.

 

[jr16]

g2*g2 = e
g2 + 2 = e
g4 = e
(g2)2 = e
Therefore, H is closed under *

does this fix the issue?

 

[HallsofIvy]

All you have done is show that g times itself is in H. That does NOT show that H is closed under multiplication.

Suppose a and b are in H. Then a^2= 1 and b^2= 1. So what can you say about (ab)^2?

By the way, as the conditions for a subgroup you give

1. H is closed under *
2. * is associative
3. The identity, e, of G is also in H
4. For each a in H, the inverse of a, a^-1, is also in H

You do NOT need to prove (2), that the operation is associative. Since that is the operation "inherited" from G you alredy know that the operation is associative from the fact that G is a group.

 

[jr16]

Thank you, all of this input is very helpful!

In my original post I should have made it more clear that * in this case does not represent multiplication but rather any binary operation on G. Therefore, sⁿ = s*s*s...n times and s^a * s^b = s^(a+b)

Let me try again:
Let a, b be elements of the set H such that a²=e and b²=e

a*b = e
a * b = a * a replacing e with a*a
b = a by the left cancellation law
?

Or should I begin with a² and b²?
so that
a² * b² = e
e * e = e
e = e
?

 

[A. Bahat]

Let a, b be elements of the set H such that a2=e and b2=e

a*b = e
a * b = a * a replacing e with a*a
b = a by the left cancellation law
?

Or should I begin with a2 and b2?
so that
a2 * b2 = e
e * e = e
e = e
?

Not quite. You have to show that if a,b are in H, then a*b is in H. You can't assume that a*b=e. Hint: consider (a*b)*(a*b) and use the fact that * is commutative.

 

[jgens]

(ii) No, G being Abelian was not a necessary condition (?)

Let G = D₆ and notice that s² = e and that (sr)²=(sr)(r^-1s) = e. But s(sr) = r and r² ≠ e.

 

[jr16]

Thank you all again for your great responses!

I had a small epiphane over breakfast this morning haha
It involves something that A. Bahat just brought up, so maybe I finally have it right this time?

Prove: a * b = e
(a * b)² = e² by squaring both sides
(a * b) * (a * b) = e * e
a * (b * a) * b = e by the associative property
a * (a * b) * a = e by the commutative property (thus, Abelian condition is necessary)
(a * a) * (b * b) = e by the associative property
(e) * (e) = e
e = e
Therefore, H is closed under *
?

 

[jgens]

Prove: a * b = e
(a * b)² = e² by squaring both sides
(a * b) * (a * b) = e * e
a * (b * a) * b = e by the associative property
a * (a * b) * a = e by the commutative property (thus, Abelian condition is necessary)
(a * a) * (b * b) = e by the associative property
(e) * (e) = e
e = e
Therefore, H is closed under *

You are not trying to prove that a,b \in H implies ab = e. You are trying to prove that if a,b \in H, then (ab)^2 = e. You can do this in essentially the same way as your work above indicates.

You should also note that just because you use the fact that G is abelian does not mean that it is necessary for the proof. It suggests that it is necessary, but you need to provide an example where the result fails if G is non-abelian in order to show that G abelian was necessary.

 

[jr16]

ah OK, let's give this another try then :)

Prove: (a*b)² = e
(a * b) * (a * b) = LS
a * (b * a) * b = LS
a * (a * b) * b = LS
(a * a) * (b * b) = LS
(e) * (e) = LS
e = LS = RS
Therefore, H is closed under *

Is this correct?

 

[jgens]

Yep. I am assuming that LS means left-hand side and RS means right-hand side.

 

[sunjin09]

There's a theorem that says subset H is a subgroup of G iff for any α, β in H, α*β^{-1} is in H. I think this would simplify your proof.