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Unformatted text preview: CSCI 124 Discrete Structures II: Groups – Deﬁnition and Examples
Poorvi L. Vora In this module, we attempt to look at the similarities between the afﬁne and shift ciphers using the notion of a group 1 A Group
with an associated operation , such that the following hold: Deﬁnition 2: A group is a set Closure is closed under the operation, i.e. Associativity The operation is associative, i.e. Identity There is an element Inverses exist Every element has an inverse in , i.e. such that . . The element is known as the identity such that . Notice that the operation is very much like addition, however, it is not always commutative. Examples of a group include: the operation; , the real numbers, with addition as the operation; the integers, with addition as the operation; matrices for any integers , the rational numbers with addition as (the real numbers without the number ) roots of one under multiplication. All ). The set of under multiplication; , , under addition; any vector space under vector addition (including, for example, the integer lattice); the complex invertible matrices is a non-abelian group under multiplication. the above examples are abelian groups, that is the operation is also commutative ( 2 An Example
with is a group, we check each of the requirements of a group: To see that Closure Suppose , i.e. and are real numbers. Their sum, is also real, i.e. . Hence the closure condition holds. 1 CSCI 124/Vora/GWU 2 Associativity Addition is associative, i.e. Identity Consider the element . Inverses exist Given inverses exist. You should similarly show that the other examples above are groups. , consider . It is a member of too, and . Hence , and . Hence the identity exists. so associativity holds. 3
For as a group under addition
, under addition . , . An example of a group is the set of remainders To see that it is a group, we check each of the requirements of a group: Closure Suppose Associativity Addition over the integers is associative, i.e. , and so associativity holds. Identity Consider the element identity exists. Inverses exist Given , consider . Hence inverses exist. The shift cipher now is as follows: . It is a member of , and , and . Hence the , and hence it is also associative . Their sum, is also in . Hence the closure condition holds. 4 as a group under multiplication ? Consider the set of remainders modulo m under multiplication. Is this a group? CSCI 124/Vora/GWU 3 Closure Suppose Associativity Multiplication over the integers is associative, i.e. associative Identity Consider the element identity exists. Inverses? Given , is that integer, , such that could be expressed as: . Such a value does not always exist. , and . Hence the , and so associativity holds. , and hence it is also . Their product, is also in . Hence the closure condition holds. If it were a group, the afﬁne cipher for where is multiplication . ...
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- Spring '09