*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.*

*Group axioms**Abelian groups**Subgroups and normal subgroups**Generators*

*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.