SubGroup

Given a GrouP G under an operation *, we say that some subset H of G is a SubGroup if H is a group under the restriction of * thereto. This occurs if and only if H is closed to products and inverses.

The subgroups of any given group form a lattice under inclusion. There is a minimal subgroup, the trivial group {e}, and a maximal subgroup, the group itself. The minimal subgroup containing some set S is denoted <S> and is said to be generated by S. Groups generated by a single element are called cyclic and are isomorphic to either the IntegerNumbers or some ModularArithmetic.

Given a subgroup H and some g in G, we define the left-coset g*H={g*h:h in H} and the right-coset H*g={h*g:h in H}. If g*H=H*g for every g in G, then H is said to be normal. In that case we can define a multiplication on cosets by

   (g1*H)*(g2*H) := (g1*g2)*H

The resulting structure is called the quotient group G/H. There is a natural homomorphism f:G -> G/H given by f(g)=g*H. The image f(H) is trivial.

In general a group HomoMorphism f: G -> K sends subgroups of G to subgroups of K. Also, the preimage of any subgroup of K is a subgroup of G. We call the preimage of the trivial group {e} in K the kernel of the homomorphism. As it turns out, the kernel is always normal and the image of G is always isomorphic to G/ker(f).

The normal subgroups of any group G form a LatticE under inclusion. The minimal and maximal elements are {e} and G, the glb of two subgroup is their intersection and their lub is a ProductGroup?. If the subgroups are not normal then the lub needn't exist.