[Tutor] determining whether a set is a group

[Sheila King]

On Sun, 29 Apr 2001 09:40:06 -0700, Sheila King <sheila@thinkspot.net>  wrote
about Re: [Tutor] determining whether a set is a group:


:If I get a better answer to "What are groups good for?", I will share.

OK, from a Britannica.com article on Fourier Analysis (which we all know is
extremely important for engineering, right?) here:
http://www.britannica.com/eb/article?eu=120670

I quote:
"""
Since their formal introduction in the early 19th century, groups have been
one of the principal objects of mathematical attention (see algebra: Groups).
Their widespread and profound applications to such physical subjects as
crystallography, quantum mechanics, and hydrodynamics and to such other
mathematical regimes as number theory, harmonic analysis, and geometry have
demonstrated their importance.
"""

And from this article at Britannica.com on Groups:
http://www.britannica.com/eb/article?eu=120647

I quote:
"""
In studying the solution of polynomial equations, a Norwegian mathematician,
Niels Henrik Abel, showed that in general the equation of fifth degree cannot
be solved by radicals. Then the French mathematician Évariste Galois, using
groups systematically, showed that the solution of an equation by radicals is
possible only if a group associated with the equation has certain specific
properties; these groups are now called solvable groups.

 The group concept is now recognized as one of the most fundamental in all of
mathematics and in many of its applications. The German mathematician Felix
Klein considered geometry to be those properties of a space left unchanged by
a certain specific group of transformations. In topology geometric entities
are considered equivalent if one can be transformed into another by an element
of a continuous group.
"""

If you're interested, you will find much more on the importance and usefulness
of groups at the links listed above.

--
Sheila King
http://www.thinkspot.net/sheila/
http://www.k12groups.org/