The symmetries of a regular tetrahedron form a group that is isomorphic to the even permutations on 4 elements. There are 12 such even permutations--exactly half of the 24 total permutations on 4 elements. We sketch a proof showing that if G is group of permutations that includes at least one odd permutation, then G contains an equal number of even and odd permutations.

This video introduces the traditional definitions of isomorphic groups and isomorphism, and illustrates how these notions can be used to show that two groups are "the same." A compelling example is explored to show the power that is achieved by using mathematical definitions rather than intuitive ones. The video presumes that the idea of isomorphic groups is already understood in an intuitive sense (in terms of Cayley graphs).

A description of the celebrated theorem of Arthur Cayley, and a proof that every group can be represented as a group of permutations. The theorem is motivated by a "magic trick" involving operation tables. Examples of the theorem are provided.

This video explores the notion of a homomorphism by using a manipulative to describe two algebraically distinct groups that are structurally similar to one another. A parallel is drawn between the algebraic notions of isomorphism versus homomorphism, and the geometric notions of isometry versus dilation.

The Fundamental Theorem of Group Homomorphisms is arguably the most important theorem in Abstract Algebra. This video proves the first part of the theorem -- that the set of domain elements that a homomorphism sends to the identity element of the range always forms a normal subgroup of the domain.

In this video we complete the proof of the Fundamental Theorem of Group Homomorphisms by showing that if K is the vanishing set of a surjective homomorphism from G1 to G2, then the quotient group G1/K is absolutely identical (algebraically) to G2. We conclude by showing that there is a one-to-one correspondence between the homomorphic images of G1, and the collection of normal subgroups of G1.

Rings are algebraic structures that have both additive multiplicative operations defined on them. The ring with the additive operation forms an abelian group while the ring with the multiplicative operation forms an associative algebra. The distributive property ties these two operations together.

This video introduces rings and discusses some of their properties and varieties.

Rings come in many flavors. The richest variety of a ring is a field. Fields are rings for which the nonzero elements form a group with respect to the multiplicative operation. It is easy to show that fields are commutative rings with unity which satisfy the cancellation property; that is, every field is an integral domain. However, the integers form an integral domain that is not a field since some (most!) integers do not have a multiplicative inverse element that is also an integer. The main result in this video shows that every FINITE integral domain is also a field.

Subrings are subsets of rings, R, that are themselves rings with respect to the addition and multiplication that are defined on R. This video shows that a subset A of a ring R is a subring of R if it is closed with respect to the addition and multiplication that are defined on R, and if A is closed with respect to additive inverses. It is noted that since the ring R is abelian with respect to addition, that every subring of R is a normal subgroup of R with respect to addition. Therefore, every subring of a ring gives rise to a quotient group of R with respect to addition. The question is raised as to which of these subrings provides enough structure to permit the cosets that comprise the elements of this quotient group to form a quotient ring in a natural way. An answer is provided to this question: if the product of elements of A with elements of R produce elements of A, then the cosets of A form a ring with respect to the natural operations of coset addition and multiplication on R/A.

If h is a surjective ring homomorphism from a ring R1 to a ring r2, then there is some set of elements of the ring R1 that gets sent to the zero element (the additive identity) of the ring R2. This set of domain elements is traditionally called the kernel of the homomorphism h. We call this set the vanishing set of h since being sent to zero is akin to vanishing. We saw in a previous video that the vanishing set of h is a subring of the domain R1. Moreover, if a is any member of the vanishing set of h and x is any member of R1, then a*x and x*a are both members of the vanishing set of h. In other words, the vanishing set K of h has the property that it absorbs products. That is, the vanishing set of h is an ideal of R1. Therefore, R1/K is a ring. We discuss in this video that R1/K is isomorphic to R2. In other words, every homomorphic image of R1 is isomorphic to a quotient ring of R1. Moreover, every quotient ring of R1 is a homomorphic image of R1. Therefore, there is a one-to-one correspondence between the homomorphic images of a ring, and the quotient rings of that ring. This result is known as the Fundamental Theorem of Ring Homomorphisms. It stands as one of the hallmarks of the study of Ring Theory.

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