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 be a finite group. The order of the group is not divisible by two. And for all we have,
Show that is abelian.
2)Let be a finite group. The order of the group is not divisible by three. And for all we have,
Show that is abelian.
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, is finite. And is infinite.
Definition: A binary operation is something that operates two elements into another element. For example, is a binary operation on as you can see . 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 is closed under because the sum of any two of these is again an element in the set. Mathematicians like to use the notation " " which means "element of". Thus, symbolically
where any two elements in and is the binary operation. For example, is not closed under (division).
It should be made clear here that is not necesarry the same as 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 which has the following property,
For any .
For example has an identity element because we have,
for any .
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, it has no inverse for because we need that,
. BUT. If we introduce the negative, then we do. Here is another example under multiplication. First we show that has an identity element, because for any . And it has an inverse for an element, all you do is flip the fraction. Thus, if you want the inverse of you flip it to get which is the inverse. Because . The important thing I said was positive. Because If I did not say that then zero has no invese :eek: .
Definition: A binary operation on a non-empty set is said to be "associative" when we have . The standard sets we look upon before are all associative. If we definie on , to be then it is associative. Because . 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 means the "inverse of " and as you guessed it means the inverse of . 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 . Rather they juxtapose the two elements
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 . That means (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 .
Funny. I just realized I explained everything except the most important term, "abelian". When we have 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 we can cancel the left elements to get . We cannot conclude that and cancel. Also, if we have again cancelation laws say . (Note: There is no such thing as division by zero in group theory. So nothing to worry about).