Bsc Mathematics Abstract Algebra by sk mapa Group Theory exercise 9 solution pdf

2.7. Finite groups.

Definition. A group (G) is said to be a finite group if G contains finite number of element The order of a finite group (G,o) is the number of elements of G. The order of the group G is denoted by o(G) or by| G|.

Theorem 1.

If (G,o) be a finite group then in every row (or column) of the composition table each element of G appears exactly once.

Group U. (Group of units modulo n).

The set Z, forms a commutative monoid under multiplication (mod n). Let us find the units in the monoid (Z)(n> 1).

Symmetric group Sn. 

Let S be the set of all permutations on the set (1,2, ...,n}.

Let us examine if S forms a group with respect to 'multiplication of

permutations'.

(i) Let f,g be two permutations on the set {1,2,...,n}. Then f.g is also a permutation on the set {1,2, ..,n}. Therefore f €S, g€S =>f.g€S.

(ii) A permutation on S is a bijective mapping from the set {1,2, ...,n} onto itself and multiplication of two permutations is the composition of two bijective mappings. Since composition of mappings is associative, multiplication of permutations is also associative.

Klein's 4-group. 1.

Let S = {e, a,b,c} and let o be the binary composition defined on S

by eoa = aoe = a, eob = boe = b, coc = coe=c,eoe=aoa=bob=coc = e, aob = boa = coc = coa = b, boc = cob=a

The composition table is given below

   o     |e.   a.    b.    c

-------- |--------------------

      e  |e.    a.    b.    c

      a  |a.    e.    c.     b

      b  |b.    c.    e.     a

      c   |c.    b.   a.     e

--------------------------------

(S,o) is an abelian group of order 4. It is called Klein's 4-group and is denoted by V. Klein was a German mathematician and V comes from

the German word Viergruppe.

An important property of the group is that every element of the group is its own inverse. 

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