Repeated elements within a group - possible?

[spaghetti3451]

Can there be repeated elements within a group?

 

[wisvuze]

I'm not entirely sure what you mean by this question.
In "reality" a group will not have "repeated elements" (whatever this means ), but there are multiple ways to write down the same element in a group. For example, in a dihedral group,

sr^-1 and rs are two ways to write down the same element in the group.

 

[algebrat]

Outline of long winded answer: after explaining why there are not really any repeats, I give two ways in which it can appear to have two of the same elements, the second way being the more interesting, so I expand on that for several paragraphs.

It is sort of fun in math to see how much structure you can ignore and still get the same result. We can ignore the group structure, and look at the underlying set. I found this at http://en.wikipedia.org/wiki/Union_(set_theory)

"Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}."

Now, when we add on the structure of a group, we can have nonidentical elements that are isomorphic. Elements are isomorphic if there is an automorphism of the group (isomorphism from the group to itself), such that the elements map to each other. Like clockwise rotation of a triangle is isomorphic to counterclockwise rotation of a triangle. The "symmetry element" has nearly indistinguishable behavior, but is not identical.

Also, two elements that can be written differently may in fact be the same, like wisvuze was pointing out. Before I realized which version of identicalness they were displaying (I thought they were giving an example of isomorphic till I wrote the previous paragraph).

Another example of that is in the formal construction of fractions, at some point it is decided that 1/2=2/4. They look like two different elements, but at some point in the construction we decided that they were equivalent. Really, when we say 1/2, we are referring to the set {n/(2n)}. The fractions are a set of sets. The elements of a quotient group are cosets.

To get a feel for the frequency of groups being a set of sets, see the standard topic of quotient groups.

Also noteworthy is generators and relations, which is one of several standard ways to represent groups. (Others are matrices and subroups of permutation groups, off the top of my head.)

To use wisvuze's example, we could say when we are at the step of generators, sr^-1 and rs are not equal. These are words in the alphabet {r,s}, and if I remember my free constructions correctly, the free group generated by them. Then, whichever relation we choose to define the dihedral group, specifies essentially how to take the quotient of the free group. Now our group is a set of sets, one of those elements is a set, which contains for instance sr^-1 and rs. In other words, these are two representatives for the same coset.

 

[wisvuze]

( a related note is the word problem for groups: given a finite set of generators and a presentation, one cannot algorithmically decide whether or not two elements "written down" represent the same element in the group [ sorry if I butchered the statement ]. Anyway, the point is, deciding whether or not two elements are the same is actually an interesting question. You can probably think of this as syntax versus semantics )

 

[HallsofIvy]

A group is, by definition, a set of objects together with an operation satisfying certain properties. A set itself does not allow the same object to be repeated so there cannot be repeated objects in a group.