1. Abstract Algebra: Subgroup Proof

Hail Mathematicians,

It's me again.

I have a problem, here is the appropriate theorem:

Theorem: Let $G$ be a group with operation $*$, and let $H$ be a subset of $G$. Then $H$ is a subgroup of $G$ iff

(a) $H$ is nonempty

(b) if $a \in H$ and $b \in H$, then $a*b \in H$, and

(c) if $a \in H$, then $a^{-1} \in H$

i've used this theorem over and over in studying, but i have trouble proving (b) and (c) in the following problem

Problem:

Assume that $G$ is a group with operation $*$ and let $Z(G) = \{ x \in G : a*x = x*a \mbox{ for every } a \in G \}$

Prove that $Z(G)$ is a subgroup of G.

This is a homework problem, so only hints please.

Thanks, guys and gals

EDIT: Proving it is nonempty is trivial, so don't even bother...

2. Originally Posted by Jhevon
Hail Mathematicians

I have a problem, here is the appropriate theorem:

Theorem: Let $G$ be a group with operation $*$, and let $H$ be a subset of $G$. Then $H$ is a subgroup of $G$ iff

(a) $H$ is nonempty

(b) if $a \in H$ and $b \in H$, then $a*b \in H$, and

(c) if $a \in H$, then $a^{-1} \in H$

i've used this theorem over and over in studying, but i have trouble proving (b) and (c) in the following problem[B]
I never used that theorem. I simply do it by definition: (i) H is a non-trivial subset of G (ii) if a*b are in H for all a,b in H (iii) e is in H (iv) if a in H then a^(-1) in H (v) * is associate (but that follows because G is a group).

Try doing it this way.

3. Originally Posted by ThePerfectHacker

I never used that theorem. I simply do it by definition: (i) H is a non-trivial subset of G (ii) if a*b are in H for all a,b in H (iii) e is in H (iv) if a in H then a^(-1) in H (v) * is associate (but that follows because G is a group).

Try doing it this way.
yes. (i) and (iii) are easy. but i can't show that a*b is in Z(G) for all a,b in Z(G).

so far i have: Let $a,b \in Z(G)$. Then for all $x \in G$ we have $a*x = x*a$ and $b*x = x*b$ ...and there i'm stuck. i need to show that $a*b \in Z(G)$

also, i need to show that for all $a \in Z(G)$ we have $a^{-1} \in Z(G)$, which is what i think you are saying in (iv)

so let $a \in Z(G)$, then for all $x \in G$ we have $a*x = x*a$ ... then what. i need to show that $a^{-1}*x = x*a^{-1}$. how do i do that?

4. does this work?

for (b):

Let $a,b \in Z(G)$. Then for all $x \in G$, we have:

$a * x = x * a$ ...............(1)

and

$b * x = x * b$ ...............(2)

Take equation (1) and apply $b$ to the both sides, we get:

$a * x * b = x * a * b$

$\Leftrightarrow a * b * x = x * a * b$ ..........by equation (2)

$\Leftrightarrow (a * b) * x = x * (a * b)$

thus, $a * b \in Z(G)$

for (c):

Let $a \in Z(G)$, and $e \in G$ be the identity element of $G$. Then for all $x \in G$ we have:

$a * x = x * a$

apply $a^{-1}$ to both sides. (we know $a^{-1}$ exists since $a \in G$, and $G$ is a group, so $a^{-1} \in G$).

$\Rightarrow a^{-1} * a * x = a^{-1} * x * a$

$\Rightarrow e * x = a^{-1} * x * a$

$\Rightarrow x = a^{-1} * x * a$

apply $a^{-1}$ again, this time to the right most position

$\Rightarrow x * a^{-1} = a^{-1} * x * a * a^{-1}$

$\Rightarrow x * a^{-1} = a^{-1} * x * e$

$\Rightarrow x * a^{-1} = a^{-1} * x$

thus, $a^{-1} \in Z(G)$

This completes the proof

Looks good?

5. By the way the subgroup Z(G) has a name, it is called the "center" of the group.

Let x and y be in the center. We ask whether xy is in the center, if and only if, for any g in G we have g(xy) = (xy)g. But that is true because g(xy) = (gx)y = (xg)y=x(gy) = (xy)g.