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 $\displaystyle G$ be a finite group. The order of the groupis notdivisible by two. And for all $\displaystyle a,b\in G$ we have,

$\displaystyle (ab)^2=(ba)^2$.

Show that $\displaystyle G$ is abelian.

2)Let $\displaystyle G$ be a finite group. The order of the groupis notdivisible by three. And for all $\displaystyle a,b\in G$ we have,

$\displaystyle (ab)^3=a^3b^3$.

Show that $\displaystyle G$ 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: Afiniteset is a set which has a finite number of elements. For example, $\displaystyle S=\{1,2,3\}$ is finite. And $\displaystyle S=\{0,1,2,...\}$ is infinite.

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

Definition: Aclosednon-empty set under some binary operation. Is a set such that the result after the binary operation is still in the set. For example $\displaystyle \{0,1,2,3,...\}$ is closed under $\displaystyle +$ because the sum of any two of these is again an element in the set. Mathematicians like to use the notation "$\displaystyle \in$" which means "element of". Thus, symbolically

$\displaystyle a*b\in S$ where $\displaystyle a,b$ any two elements in $\displaystyle S$ and $\displaystyle *$ is the binary operation. For example, $\displaystyle T=\{1,2,3,...\}$is notclosed under $\displaystyle /$ (division).

It should be made clear here that $\displaystyle a*b$ is not necesarry the same as $\displaystyle b*a$ 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 $\displaystyle e$ which has the following property,

$\displaystyle a*e=e*a=a$

For any $\displaystyle a\in S$.

For example $\displaystyle N=\{0,1,2,3,...\}$ has an identity element $\displaystyle 0$ because we have,

$\displaystyle x+0=0+x=x$ for any $\displaystyle x\in N$.

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, $\displaystyle N=\{0,1,2,3,...\}$ it has no inverse for $\displaystyle 1\in N$ because we need that,

$\displaystyle 1+x=x+1=0$. BUT. If we introduce the negative, $\displaystyle N'=\{-3,-2,-1,0,1,2,...\}$ then we do. Here is another example $\displaystyle Q=\{\mbox{positive rationals}\}$ under multiplication. First we show that $\displaystyle Q$ has an identity element, because $\displaystyle 1*x=x*1=x$ for any $\displaystyle x\in Q$. And it has an inverse for an element, all you do is flip the fraction. Thus, if you want the inverse of $\displaystyle 1/2$ you flip it to get $\displaystyle 2/1$ which is the inverse. Because $\displaystyle (2/1)*(1/2)=(1/2)*(2/1)=1$. 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 $\displaystyle a*(b*c)=(a*b)*c$. The standard sets we look upon before are all associative. If we definie $\displaystyle *$ on $\displaystyle N=\{0,1,2,...\}$, to be $\displaystyle a*b=a$ then it is associative. Because $\displaystyle 1*(2*3)=(1*2)*3$. 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 $\displaystyle *$ is a "group" when it is: closed, associative, has identiy element, for each element has inverse.

Just to mention some notation $\displaystyle a^{-1}$ means the "inverse of $\displaystyle a$" and as you guessed it means the inverse of $\displaystyle a$. 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 $\displaystyle a*b$. Rather they juxtapose the two elements $\displaystyle ab$

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 $\displaystyle a^n=e$. That means $\displaystyle \underbrace{aaa....a}_n=e$ (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 $\displaystyle (ab)^{-1}=b^{-1}a^{-1}$.

Funny. I just realized I explained everything except the most important term, "abelian". When we have $\displaystyle a*b=b*a$ 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 $\displaystyle ax=ay$ we can cancel the left elements to get $\displaystyle x=y$. We cannot conclude that $\displaystyle ax=ya$ and cancel. Also, if we have $\displaystyle xa=ya$ again cancelation laws say $\displaystyle x=y$. (Note: There is no such thing as division by zero in group theory. So nothing to worry about).