A square has eight symmetries. These are:. These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry.

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Composing two of these symmetry functions gives another symmetry function. These symmetries determine a group called the dihedral group of degree 4, denoted D 4. The underlying set of the group is the above set of symmetry functions, and the group operation is function composition. The result of performing first a and then b is written symbolically from right to left as. The group table on the right lists the results of all such compositions possible. Using the above symbols, highlighted in blue in the group table:. Given this set of symmetries and the described operation, the group axioms can be understood as follows:.

In other words, D 4 is not abelian, which makes the group structure more difficult than the integers introduced first. Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory.

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Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. The axioms may be weakened to assert only the existence of a left identity and left inverses.

## Abstract Algebra: A Comprehensive Treatment

Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above. Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element. The term b on the first line above and the c on the last are equal, since they are connected by a chain of equalities. In other words, there is only one inverse element of a.

Similarly, to prove that the identity element of a group is unique, assume G is a group with two identity elements e and f. In this case, the group operation is often denoted as an addition , and one talks of subtraction and difference instead of division and quotient. A consequence of this is that multiplication by a group element g is a bijection. This function is called the left translation by g. If G is abelian, the left and the right translation by a group element are the same. To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.

This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups.

The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category , in this case the category of groups. In other words, the result is the same when performing the group operation after or before applying the map a. Thus a group homomorphism respects all the structure of G provided by the group axioms. From an abstract point of view, isomorphic groups carry the same information. Informally, a subgroup is a group H contained within a bigger one, G.

Knowing the subgroups is important in understanding the group as a whole. Given any subset S of a group G , the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S.

## Abstract Algebra: A Comprehensive Treatment - Claudia Menini, Freddy Van Oystaeyen - Google Books

Again, this is a subgroup, because combining any two of these four elements or their inverses which are, in this particular case, these same elements yields an element of this subgroup. In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup.

For example, in D 4 above, once a reflection is performed, the square never gets back to the r 2 configuration by just applying the rotation operations and no further reflections , i. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are.

The left cosets of any subgroup H form a partition of G ; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection.

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Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i. In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or factor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N , the quotient group is defined by. The group operation on the quotient is shown at the right. Quotient groups and subgroups together form a way of describing every group by its presentation : any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations.

Together with the relations. A presentation of a group can also be used to construct the Cayley graph , a device used to graphically capture discrete groups. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon. Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above.

If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups.

The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic and thus irrelevant , because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point.

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The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop or any other loop winding once around the hole. This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory , in particular when implemented for finite groups.

Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules , vector spaces and algebras also form groups.

The desire for the existence of multiplicative inverses suggests considering fractions. Fractions of integers with b nonzero are known as rational numbers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. The rational numbers including 0 also form a group under addition. Group theoretic arguments therefore underlie parts of the theory of those entities. In modular arithmetic , two integers are added and then the sum is divided by a positive integer called the modulus.

The result of modular addition is the remainder of that division. This is familiar from the addition of hours on the face of a clock : if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. For any prime number p , there is also the multiplicative group of integers modulo p.

The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication. Finally, the inverse element axiom requires that given an integer a not divisible by p , there exists an integer b such that.

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Hence all group axioms are fulfilled. They are crucial to public-key cryptography. A cyclic group is a group all of whose elements are powers of a particular element a. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as.

Indeed, each element is expressible as a sum all of whose terms are 1.