Introductory Theorems on Group Theory:
A group (G,*) is an ordered pair consisting of a set G (refered to as a group with respect to *) and a
binary operation * : G X G => G (called the group operation) for which the following properties hold:
I) For any
Now let f be any permutation in Sym S. Now f is a composition of transpositions, as kn . k2 k1. If f
is a composition of an odd (resp. even) number of transpositions, then f is said to be odd (resp. even).
Lemma:
Several properties of parity are readily s
elements, then Sym S contains n! functions.
Sym S, called the symmetric group on S, is a group with respect to the operation of function
composition: If f and g are in S, then the function g f (read as "g compose f") is also in S, since the
composition of
Notice that cfw_E,O, which we defined as some mysterious set with binary operation *, is actually the
same thing as Z2! O acts exactly like the element 1, and E acts exactly like 0, and * acts exactly like
addition. Not surprisingly, Z2 is isomorphic to c
Let the order of g be t. Then f(g)^t = f(g).f(g) = f(g.g) = f(g^t) = f(e) = e. So f(g)^t does equal the
identity element of H. This does not prove that t is the order of f(g)^t-there could be some smaller
power of f(g) also equal to the identity element o
(I) qH contains n distinct elements. For if any two elements in qH were the same, as q*gi = q*gj
(where gi, gj are in H), then canceling we would have gi = gj, which is contrary to the supposition that
all the elements in H are different.
(II) qH has no e
that g, g^2, g^3, etc. are all distinct elements (if a^i = a^j where i, j are unequal and j then canceling we
get a^i-j = e, contrary to the supposition that a^t is the smallest power equal to e).
Furthermore, H = cfw_e, g, g^2, . g^t-1-i.e. any power of
that g, g^2, g^3, etc. are all distinct elements (if a^i = a^j where i, j are unequal and j then canceling we
get a^i-j = e, contrary to the supposition that a^t is the smallest power equal to e).
Furthermore, H = cfw_e, g, g^2, . g^t-1-i.e. any power of
congruent to." Just as we have the laws of identity, symmetry, and transitivity for all objects, so too do
we have identity, symmetry, and transitivity for isomorphic groups.
An important structure related to a group is a subgroup. Specifically, if G is s
congruent to." Just as we have the laws of identity, symmetry, and transitivity for all objects, so too do
we have identity, symmetry, and transitivity for isomorphic groups.
An important structure related to a group is a subgroup. Specifically, if G is s
g^-n is defined as g^-1*.g^-1 (n times) and g^0 is defined as the identity element e.
It is easy to check that these terms are well defined and that the properties of exponents hold as usual-specifically that g^a*g^b = g^(a+b) for any integers a, b.
Defin
g^-n is defined as g^-1*.g^-1 (n times) and g^0 is defined as the identity element e.
It is easy to check that these terms are well defined and that the properties of exponents hold as usual-specifically that g^a*g^b = g^(a+b) for any integers a, b.
Defin
Introductory Theorems on Group Theory:
A group (G,*) is an ordered pair consisting of a set G (refered to as a group with respect to *) and a
binary operation * : G X G => G (called the group operation) for which the following properties hold:
I) For any
Lecture 6:
(Introductory definitions and theorems of group theory)
We are introduced to Cayley Tables, a way of defining a binary operation on a set.
Consider for example the table
a|b|c
a|a a c
b|a b c
c|c c b
This table shows that a*a = a, c*c = b, etc.