List four modes of radioactive decay that results in the transmutation of one element into another?

gamma?

A nuclear transmutation occurs when one element is converted into another element through a spontaneous emission of particles from the nucleus or by induced emission of particles by bombardment of a nucleus by particles like neutrons, protons, or other nuclei from atoms.

As you mention, alpha and beta decay, and fission occur to cause transmutation.  Another form of transmutation occurs in our atmosphere.  Cosmic rays, ultra high energy photons, collide with our atmosphere interacting with N-14 producing C-14.  This is the primary method for the introduction of C-14 into our environment.

Another form of transmutation occurs when elements are bombarded by neutrons.  Neutron bombardment is a commonly used technique for producing another element.

Hope this is helpful to you.

the 4th mode is k-capture, also called electron capture. an electron from the K orbit is captured by the nucleus. see this link
http://en.wikipedia.org/wiki/K_capture