Basic Facts from Group Theory

1 Basic Facts from Group Theory Group Order 

Thus if the order of G is prime, G only has the trivial subgroups G and 1 (where G is written multiplicatively). 

The powers of g form a subgroup of G, denoted by g , with kg l = ord(g). The element g is called a generator of G if g = G. A group that has a generator is called cyclic. 

The order of any element g divides the group order. In particular, = 1 for all g e IGI this is Fermat s little theorem. 

Any cyclic group G is isomorphic to the additive group of integers modulo IGI. For any generator g, the exponentiation function expgix) = g is an isomorphism into G. The inverse is the discrete-logarithm function logg. In particular, G is Abelian, i.e., commutative. 

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The complexity of the system implies that many phenomena are not directly explainable by the basic theories of semiconductor electrochemistry. The basic theories are developed for idealized situations, but the electrode behavior of a specific system is almost always deviated from the idealized situations in many different ways. Also, the complex details of each phenomenon are associated with all the processes at the silicon/electrolyte interface from a macro scale to the atomic scale such that the rich details are lost when simplifications are made in developing theories. Additionally, most theories are developed based on the data that are from a limited domain in the multidimensional space of numerous variables. As a result, in general such theories are valid only within this domain of the variable space but are inconsistent with the data outside this domain. In fact, the specific theories developed by different research groups on the various phenomena of silicon electrodes are often inconsistent with each other. In this respect, this book had the opportunity to have the space and scope to assemble the data and to review the discrete theories in a global perspective. In a number of cases, this exercise resulted in more complete physical schemes for the mechanisms of the electrode phenomena, such as current oscillation, growth of anodic oxide, anisotropic etching, and formation of porous silicon. 

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