Fundamentals of Group Theory provides an advanced look at the basic
theory of groups. Standard topics in the field are covered alongside a
great deal of unique content. There is an emphasis on universality when
discussing the isomorphism theorems, quotient groups and free groups as
well as a focus on the role of applying certain operations, such as
intersection, lifting and quotient to a "group extension". Certain
concepts, such as subnormality, group actions and chain conditions are
introduced perhaps a bit earlier than in other texts at this level, in
the hopes that the reader would acclimate to these concepts earlier.
Some additional features of the work include:
- An historical look at how Galois viewed groups.
- The problem of whether the commutator subgroup of a group is the same
as the set of commutators of the group, including an example of when
this is not the case.
- The subnormal join property, that is, the property that the join of
two subnormal subgroups is subnormal.
- Cancellation in direct sums.
- A complete proof of the theorem of Baer characterizing nonabelian
groups with the property that all of their subgroups are normal.
- A somewhat more in depth discussion of the structure of p-groups,
including the nature of conjugates in a p-group, a proof that a p-group
with a unique subgroup of any order must be either cyclic (for p>2) or
else cyclic or generalized quaternion (for p=2) and the nature of groups
of order p^n that have elements of order p^(n-1).
- A discussion of the Sylow subgroups of the symmetric group in terms
of wreath products.
- An introduction to the techniques used to characterize finite simple
groups.
- Birkhoff's theorem on equational classes and relative freeness.
This book is suitable for a graduate class in group theory, part of a
graduate class in abstract algebra or for independent study. It can also
be read by advanced undergraduates. The book assumes no specific
background in group theory, but does assume some level of mathematical
sophistication on the part of the reader.
