Index Concordia

Group theory – Week 02

Posted in Group theory by Index Guy on September 17, 2007

During this (past) short week we mentioned some more results and definitions from finite group theory.

We started with the Dihedral group D_{2n}, which is the group of symmetries of a n-sided polygon. It consists of n rotations and n reflections. The idea is to have the polygon laying on a plane and paint one side blue and the other one red (or whichever colors you prefer) in order to distinguish sides. This group has 2n elements.

Next we came upon Lagrange’s theorem for finite groups:

  • The order h of a subgroup H of a finite group G is a divisor of the order of G.

The order of a group is the number of elements. Meanwhile, the order of an element is defined as the power t such that a^t = e. It turns out that the set of elements

\left\{e, a, a^2, ..., a^{t-1}\right\},

forms a group, the Cyclic group C_{t}. If G has as order a prime number p then the order of any element in G is p.

We can also take direct products of two groups, defined as the set

G \otimes H with elements g_{i}h_{j}

and requiring that elements of different groups commute. The direct product of two groups is also a group. For example, the Klein group is isomorphic to C_{2} \otimes C_{2}.

Besides this we also have the set of all the permutations of n objects. This forms a group of n! elements called the, guess what, permutation group S_{n}. From Cayley we have a theorem concerning finite groups and permutation groups:

  • Every finite group G of order nis a subgroup of the permutation group S_{n}.

Then we proceeded to list of finite groups, but we got tired by order 8… At this order we have the Quaternions, which are kind of interesting.

We defined invariant subgroups, also known as normal, as a subgroup N whose elements obey ana^{-1} \in N for any element in the higher group. When a subgroup is normal, its left cosets are equal to its right cosets. More definitions: a homomorphism is a map \varphi between groups that preserve the group multiplication laws. The kernel of a map \varphi is the set of all the elements in the domain that get mapped to the identity of the range. It turns out that

  • The kernel of a homomorphism is a normal subgroup.

Finally we came upon quotient groups. We start with a group G and an invariant subgroup N. Then we form the set

\left\{N, aN, bN, ...\right\}.

This forms a group G/N under the operation

(aN)(bN) = (ab)N.

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