³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980396440³2ip³2129.97.112.15³2data³2newauthor³31³3minor³30³3text³3Given 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.

A group HomoMorphism f:G->K sends subgroups of G to subgroups of K.  Moreover, the preimage of any subgroup of K is a subgroup of G.  Of particular importance is the preimage of {e}, called the kernel of the homomorphism.  Any kernel H always satisfies the property g*H=H*g for all g in G.  Such subgroups are called normal.  There is a homomorphism f with any given normal subgroup H as its kernel, and moreover the homomorphic images under all such f are isomorphic to the QuotientGroup G/H.
³3summary³3*³2tscreate³2980396440³2keepts³2980397515³2id³21076³2host³2cnts1p15.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980397515³2ip³2129.97.112.15³2data³2text³3Given 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).
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980396440³2keepts³2980397525³2id³21076³2host³2cnts1p15.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980397525³2ip³2129.97.112.15³2data³2newauthor³30³3minor³31³3text³3Given 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).
³3summary³3*³2tscreate³2980396440³2keepts³2980442334³2id³21076³2host³2cnts1p15.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980442334³2ip³2129.97.130.9³2data³2text³3Given 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.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980396440³2keepts³2980497013³2id³21076³2host³2cnts3p09.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980497013³2ip³2129.97.130.42³2data³2newauthor³30³3minor³31³3text³3Given 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.
³3summary³3*³2tscreate³2980396440³2keepts³2980717868³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³25