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A group is called Abelian if the operation * is commutative, i.e. if a * b = b * a for all a and b in G. In this case, the operation is often written as + instead of *, and the inverse of the element a is written as (-a). |
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A group is called Abelian if the operation * is commutative, i.e. if a * b = b * a for all a and b in G. In this case, the operation is often written as + instead of *, the identity element as 0, and the inverse of the element a is written as (-a). |
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Some important concepts: subgroups, permutation groups and group actions. Some examples of groups with additional structure are rings, modules and Lie groups. |
Symmetry groups |
Symmetry groups |
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Initially, one needs tools to compare groups (group homomorphisms) and to construct new groups from old ones (subgroups, normal subgroups, factor groups, center of a group, derived group, and product of groups, especially semidirect? and direct product). When studying these concepts, one encounters the theorem of Lagrange, the fundamental theorem on homomorphisms, and the [isomorphism theorems]?. An important tool here is the concept of a coset (see under subgroup). |
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Initially, one needs tools to compare groups (group homomorphisms) and to construct new groups from old ones (subgroups, normal subgroups, factor groups, center of a group, derived group, and product of groups, especially semidirect? and direct product). When studying these concepts, one encounters the theorem of Lagrange, the fundamental theorem on homomorphisms, and the [isomorphism theorems]?. An important tool here is the concept of a coset of a subgroup (see under subgroup). For more detailed study of the lattice of subgroups of a given finite group, the notion of p-group and the Sylow theorems are useful. A helpful tool for proving these theorems is the concept of a group action. |
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For more detailed study of the lattice of subgroups of a given finite group, the notion of p-group and the Sylow theorems are useful. |
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The [cyclic groups]?, groups that can be generated by a single element, can be completely characterized; they are all abelian. More generally, all finitely generated (and in particular the finite) abelian groups can be completely classified, a theorem which finds wide applications. (see [finitely generated abelian group]?). |
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The finitely generated (and in particular the finite) abelian groups can be completely classified (see [finitely generated abelian group]?). The situation is much more complicated when trying to get a handle on all finite groups. Every finite group is built up from [simple group]?s, and in a celebrated huge theorem, all finite simple groups have at last been classified (see classification of finite simple groups). |
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The situation is much more complicated when trying to get a handle on the finite non-abelians groups. Every finite group is built up from [simple groups]?, and in a celebrated huge theorem, all finite simple groups have at last been classified (see classification of finite simple groups). |
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An important tool in group theory are group representations; one basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study. |
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An important tool in group theory are group representations; one basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study. |
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Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of [automorphism group]?s. |
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Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of [automorphism groups]?. |
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Abelian groups underly several other structures that are studied in abstract algebra, such as rings, fields?, and modules. |
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