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ThePerfectHacker · #1 December 18th, 2006, 08:14 AM

Here are two questions on group theory. Do not worry this is the most elementary group theory, so nothing complicated. Look below, I made a tutorial that explains what a group is in the most simple terms!

1)Let $G$ be a finite group. The order of the group is not divisible by two. And for all $a,b\in G$ we have,
$(ab)^2=(ba)^2$ .
Show that $G$ is abelian.

2)Let $G$ be a finite group. The order of the group is not divisible by three. And for all $a,b\in G$ we have,
$(ab)^3=a^3b^3$ .
Show that $G$ is abelian.
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Hacker's Guide to Group Theory.

Here is a really simple intro (which will be mathematically limited) to such a beautiful theory.

I presume you heard the term "set". If you did not then GET OUT OF THE FORUM. (It is a collection of something, called elements. Like the set of all positive integers.)

Definition: A finite set is a set which has a finite number of elements. For example, $S=\{1,2,3\}$ is finite. And $S=\{0,1,2,...\}$ is infinite.

Definition: A binary operation is something that operates two elements into another element. For example, $+$ is a binary operation on $\{0,1,2,...\}$ as you can see $1+2=3, 2+3=5, 6+9=15,...$ . It takes two elements and transfroms them into another element.

Definition: A closed non-empty set under some binary operation. Is a set such that the result after the binary operation is still in the set. For example $\{0,1,2,3,...\}$ is closed under $+$ because the sum of any two of these is again an element in the set. Mathematicians like to use the notation " $\in$ " which means "element of". Thus, symbolically
$a*b\in S$ where $a,b$ any two elements in $S$ and $*$ is the binary operation. For example, $T=\{1,2,3,...\}$ is not closed under $/$ (division).

It should be made clear here that $a*b$ is not necesarry the same as $b*a$ as in our division example just above.

Definition: An "identity element" in a non-empty set with a binary operation such that has an element $e$ which has the following property,
$a*e=e*a=a$
For any $a\in S$ .
For example $N=\{0,1,2,3,...\}$ has an identity element $0$ because we have,
$x+0=0+x=x$ for any $x\in N$ .

Definition: An "inverse of an element" in a set with an identity element is an element that when operated with this element both ways returns back the identity element.
For example, $N=\{0,1,2,3,...\}$ it has no inverse for $1\in N$ because we need that,
$1+x=x+1=0$ . BUT. If we introduce the negative, $N'=\{-3,-2,-1,0,1,2,...\}$ then we do. Here is another example $Q=\{\mbox{positive rationals}\}$ under multiplication. First we show that $Q$ has an identity element, because $1*x=x*1=x$ for any $x\in Q$ . And it has an inverse for an element, all you do is flip the fraction. Thus, if you want the inverse of $1/2$ you flip it to get $2/1$ which is the inverse. Because $(2/1)*(1/2)=(1/2)*(2/1)=1$ . The important thing I said was positive. Because If I did not say that then zero has no invese

.

Definition: A binary operation on a non-empty set is said to be "associative" when we have $a*(b*c)=(a*b)*c$ . The standard sets we look upon before are all associative. If we definie $*$ on $N=\{0,1,2,...\}$ , to be $a*b=a$ then it is associative. Because $1*(2*3)=(1*2)*3$ . It is unusual to find an non-associative example, because they are one of the most important properties in a binary operation.

Definition: A non-empty set with some binary operation $*$ is a "group" when it is: closed, associative, has identiy element, for each element has inverse.

Just to mention some notation $a^{-1}$ means the "inverse of $a$ " and as you guessed it means the inverse of $a$ . It happens to be that the inverse is unique, that there is no ambiguity in writing this, mathematicians say, "well-defined".

I would also like to mention. That mathematicians are lazy and they do not want to waste time writing $a*b$ . Rather they juxtapose the two elements $ab$

Here are some important theorems, that you might use to prove the above problem.

Definition: The "order of an element" (if it exists) in a group is the smallest positive integer such that $a^n=e$ . That means $\underbrace{aaa....a}_n=e$ (identity element). The "order of a group" (finite) is the number of elements in it.

Theorem: In a finite group the order of any element divides the order of the group!

Theorem: In a group $(ab)^{-1}=b^{-1}a^{-1}$ .

Funny. I just realized I explained everything except the most important term, "abelian". When we have $a*b=b*a$ this is called commutative. Thus, a commutative group is called "abelian". After a great mathematician Neils Henrik Abel, who made fundamental work on commutative groups.
Now you know enough to show this.

There is another important rule, I forgot to mention. Sorry.
Theorem: In a group we have "left-right cancellation laws". Meaning that if $ax=ay$ we can cancel the left elements to get $x=y$ . We cannot conclude that $ax=ya$ and cancel. Also, if we have $xa=ya$ again cancelation laws say $x=y$ . (Note: There is no such thing as division by zero in group theory. So nothing to worry about).