Mathematicians who are algebraists (algebraist is just a fancy name for a mathematician who specializes in one of the many different fields of mathematics that just happens to have the name algebra) all are very familiar with mathematical structures called groups, rings, and fields. Groups, rings and fields are closely related; groups just happen to have the least number of axioms that describe them. (An axiom is a mathematical proposition that states a generally accepted truth.) What follows on this page, is a short tutorial on the nature of groups. If you are already a mathematician, you will be terribly bored by all of this. If you are not, you will need this information to understand what is said elsewhere in this site.
A group is a set of objects and an operation on these objects, normally called multiplication, that satisfies the following conditions.
1. A group is closed under the defined operation.
This just means that if two objects a and b are members of a group, then ab = c is also in the group.
2. The operation must be associative.
This means that for a, b, and c that are members of a group then a(bc) = (ab)c.
3. A group must have an identity element.
This means that there must be some element i in a group such that for any other element a, also in the group,
ai = ia = a.
4. A group contains inverses.
This means that for every element a in a group, there must exist an element j such that aj = ja = i, the identity element.
Any set that satisfies all of these axioms is called a group.
In general, the elements of a group do not commute with one another. That is, if a and b are elements of some group, then ab is not necessarily equal to ba. If a some group G has the property that for any a and b in G, ab = ba, we call that group an abelian group. (Niels Henrik Abel was a ninteenth century mathematician. For a biography of Abel go to http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html)
A subgroup is a collection of elements from a group. All the elements in a subgroup must be members of the group. Also a subgroup must satisfy all of the group axioms. That is, for any group G, and H a subgroup of G, if a and b are members of H, then ab and ba must also be members of H (closed), i must be in H, H must be associative, and for every element a in H, the inverse of a must also be in H. In other words, H is also a group. All of the elements in H just happen to also be in G.
Before I explain what a normal subgroup is I need to talk about cosets. A coset of a subgroup is the set that results when every element in a subgroup is multiplied by any unique element of the larger group. Since groups do not generally commute, there are left cosets and right cosets. Generally, if G is a group and H is a subgroup of G, then a left coset of H is formed by taking any element g in G and multiplying each element h in H by g from the left side. Similarly, a right coset would be formed by multiplying every h by g from the right side. We usually denote a left coset of H in G by gH and a right coset of H in G by Hg.
H is a normal subgroup of G if and only if for every g in G, gH = Hg. That is, H in a normal subgroup of G if and only if every right coset of H in G is a left coset of H in G.
Normal subgroups are very important in group theory. Not all groups have normal subgroups. Also if H, a subgroup of G is abelian, then H is always a normal subgroup of G.
A generator set of a group is a subset of members of the group from which the entire group may be "generated" by products of elements of the subset. A generator set is subgroup. It can be the whole group. A minimal generator set is the generator set with the least number of elements needed to generate the whole group. If the minimal generator set consists of only the identity and one other element, the group is called a cyclic group.
Now we are ready to talk about the structure of wallpaper groups. To continue, select one of the options below.
Go to Isometries of the Euclidian Plane.
References:
Herstein, I. N. (1975). Topics in Algebra. New York, John Wiley and Sons.
Allenby, R. B. J. T. (1991). Rings, fields and groups: An introduction to Abstract Algebra. London, Edward Arnold.
This page was created on April 21, 2001 by Michael E. McCallum. For information or comments please contact mmccallu@bellsouth.net.