Classification of Groups
Groups have a wide range at its fully expanded level. This range may include finite permutation groups and matrix groups to abstract groups. These groups are likely to be specified on the basis of their presentation by their very own generators and relations. One needs to get familiar with following main classification groups which are more important according to mathematical studies than other minor classes of groups.
Permutation Groups:
Permutation group was the very first class of groups that underwent through systematic studies. Consider X is a given set and G is a collection of bijections of X. G is enclosed under compositions and inverses. Here G is a group that is acting on set X. if X consists of ‘n’ elements and G contains every permutations then G is said to be a Symmetric group. Generally, G is a subgroup of symmetric group of group X which contains permutations. Because of Cayley exhibited any group as a permutation group, that is actually acting upon itself (indicated as X=G) in order to keep regular representation.
In certain situations, the architecture of a permutation group is good enough to be studied or observed using the characteristics of its action on the related set. For instance, following this approach one proves that n>=5, and the alternating group denoted with An simple. It is clear that any proper normal subgroups are no admitted by it. This idea is considered as a key fact and plays a crucial role in the impossibility of solving a general equation of degree nl >=5 in radicals.
Matrix Groups:
Matrix groups are another important class of groups. They are also knows as linear groups. Here G is assumed as a set that consists of invertible matrices that are sorted in ‘n’ order and a field indicated with K that is closed under the product and inverses. This kind of group acts on the n-dimensional vector space which is indicated by Kn. It performs its task with the help of linear transformations. This action is a conceptual similarity between matrix groups and permutation groups. The geometry used in the action is actually used to construct properties of the group G.
Transformation Group:
Permutation and matrix groups are special cases of this particular group. Transformation groups are those groups which act on a certain space X, meanwhile preserving its inherent structure. For permutation groups, X is and in case of matrix X is considered as a vector space. The concept of transformation group tends to be closely related with the concept of symmetry group. Transformation group constantly consist all those transformation in which certain structure is preserved.
The concept of transformation group is an important class which plays a huge role in constructing a connection or link between Group Theory and differential geometry. Lie and Klein conducted a long term research, in which they consider group actions on the basis of homeomorphisms or diffeomorphisms manifolds. These Matrix groups can be discrete or continuous themselvesAbstract Groups:
In the beginning stage in the process of the developing group theory, most of the groups were considered “concrete”. They were realized through numbers, permutation and as well as matrices. This situation was altered in late 19th century, when concept of abstract group was being developed. The operations of abstract group tend to satisfy certain axioms systems. To specify an abstract group, presentation of generators and relations is used.
It is presented through following formula:
G= (S|R).
Quotient group is a factor group that is used to extract multiple resources in field of abstract groups. Some mathematicians consider algebraic number fields as the earliest example of factor group, a well known branch of group theory. Many concepts of number theory tend to show much interest in this concept. For example:Consider that permutations of any given set X are stored in group G, which forms a factor group that is denoted as G|H. This factor group will no longer act on set X. As a result to changes occurred in concrete and abstract group, a requirement rises for consideration of group properties. It helps to establish a result which is applied on the whole class of group, rather than trying to acquire result for a certain group. This change is of great importance for the development of advanced mathematics, as it tends to provide a solid base for construction of algebra at abstract level, in simple words it is abstract algebra.
Topological and Algebraic Groups:
When group G is endowed with extra structure it provokes elaboration of the concept of group and it tends to be an important elaboration. Here, one should notice that G is endowed with additional structures such as topological space, differentiable manifold and algebraic variety.
M: G x G, (g, h) -> gh, i: G->G, g->g-1,
If a certain group operation ‘m’ that represents multiplication and ‘i’ (for inversion) is compatible with the structure that is mentioned above, then G is said to be a topological, Lie or algebraic group. The group that follows the structure mentioned above is usually smooth, regular and continuous.
The addition of extra structures in G brings up a relation between these groups and various types of mathematical concepts. It means that it makes more tools or let’s says ideas become available in their studies. A natural domain is formed by Topological groups for abstract harmonic analysis. Lie groups those are also realized as transformation groups. They are mainstays of representation theory and differential geometry. Complete classification is achieved for lie groups connected with compacts. There is a considerable connection between abstract groups and topological groups: that is whenever a certain group is able to be realized as lattice that is a component of topological group G, in such case geometry to G generates important results about that group. In recent trend of this theory, finite groups are exploited with their relation with topological groups, especially compact one. For instance, consider G as a single analytic group that has family of quotations. Same are finite groups of various orders. Here properties of G translate into the properties of its finite quotients.
Would you like to read more about this topic? This book might interest you: Group Theory.