{"id":7905,"date":"2022-05-11T14:28:16","date_gmt":"2022-05-11T08:58:16","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7905"},"modified":"2026-04-24T16:34:26","modified_gmt":"2026-04-24T11:04:26","slug":"what-is-a-group","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-a-group\/","title":{"rendered":"What is a group?"},"content":{"rendered":"\n

A group<\/a> is an algebraic structure (G, *) in which the binary operation * on G satisfies the following conditions:<\/p>\n\n\n\n

Condition-1<\/strong>: For all a, b, c, \u2208 G<\/p>\n\n\n\n

a* (b * c) = (a * b) * c (associativity)<\/p>\n\n\n\n

Condition-2: <\/strong>There exists an elements e \u2208G such that for any a \u2208G<\/p>\n\n\n\n

a* e= e * a = a (existence of identity)<\/p>\n\n\n\n

Condition-3:<\/strong> For every a \u2208G, there exists an element denoted by in G such that<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

is called the inverse of a in G.<\/p>\n\n\n\n

is called the inverse of a in G.<\/p>\n\n\n\n

Example: (Z, +) is a group where Z denote the set of integers.<\/strong><\/p>\n\n\n\n

Example: (R, +) is a group where R denote the set of real numbers.<\/strong><\/p>\n\n\n\n

Different types of groups<\/h2>\n\n\n\n

Abelian group<\/strong><\/p>\n\n\n\n

Let (G, *) be a group. If * is commutative that is<\/p>\n\n\n\n

a* b = b * a for all a, b \u2208G then (G, *) is called an Abelian.<\/p>\n\n\n\n

Finite group-<\/strong><\/p>\n\n\n\n

G is said to be a finite group, if the set G is a finite set.<\/p>\n\n\n\n

Infinite group-<\/strong><\/p>\n\n\n\n

A group G, which is not finite is called an infinite group.<\/p>\n\n\n\n

Order of a finite group<\/strong><\/h2>\n\n\n\n

The order of a finite group (G, *) is the number of distinct element in G. The order of<\/p>\n\n\n\n

G is denoted by O (G) or by |G|.<\/p>\n\n\n\n

Example<\/strong><\/h2>\n\n\n\n

If G = {1, -1, i, -i} where , then  show that G is an abelian group with respect to multiplication as a binary operation.<\/strong><\/p>\n\n\n\n

Sol.<\/p>\n\n\n\n

First we will construct a composition table-<\/p>\n\n\n\n

.<\/strong><\/td>1<\/strong><\/td>-1<\/strong><\/td>I<\/strong><\/td>-i<\/strong><\/td><\/tr>
1<\/strong><\/td>1<\/strong><\/td>-1<\/strong><\/td>I<\/strong><\/td>-i<\/strong><\/td><\/tr>
-1<\/strong><\/td>-1<\/strong><\/td>1<\/strong><\/td>-i<\/strong><\/td>i<\/strong><\/td><\/tr>
i<\/strong><\/td>i<\/strong><\/td>-i<\/strong><\/td>-1<\/strong><\/td>1<\/strong><\/td><\/tr>
-i<\/strong><\/td>-i<\/strong><\/td>I<\/strong><\/td>1<\/strong><\/td>-1<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

It is clear from the above table that algebraic structure (G, .) is closed and satisfies the following conditions.<\/p>\n\n\n\n

Associativity- For any three elements a, b, c \u2208G (a \u22c5b) \u22c5c = a \u22c5(b \u22c5c)<\/p>\n\n\n\n

Since<\/p>\n\n\n\n

1 \u22c5(\u22121 \u22c5i) = 1 \u22c5\u2212i= \u2212i<\/p>\n\n\n\n

(1 \u22c5\u22121) \u22c5i= \u22121 \u22c5i= \u2212i<\/p>\n\n\n\n

\u21d21 \u22c5(\u22121 \u22c5i) = (1 \u22c5\u22121) i<\/p>\n\n\n\n

Similarly, with any other three elements of G the properties hold.<\/p>\n\n\n\n

\u2234 Associative law holds in (G, \u22c5)<\/p>\n\n\n\n

Existence of identity: 1 is the identity element (G, \u22c5) such that 1 \u22c5a = a = a \u22c51 \u2200a \u2208G<\/p>\n\n\n\n

Existence of inverse: 1 \u22c51 = 1 = 1 \u22c51 \u21d21 is inverse of 1<\/p>\n\n\n\n

(\u22121) \u22c5(\u22121) = 1 = (\u22121) \u22c5(\u22121) \u21d2\u20131 is the inverse of (\u20131)<\/p>\n\n\n\n

i\u22c5(\u2212i) = 1 = \u2212i\u22c5i\u21d2\u2013iis the inverse of iin G.<\/p>\n\n\n\n

\u2212i\u22c5i= 1 = i\u22c5(\u2212i) \u21d2iis the inverse of \u2013iin G.<\/p>\n\n\n\n

Hence the inverse of every element in G exists.<\/p>\n\n\n\n

Thus all the axioms of a group are satisfied.<\/p>\n\n\n\n

Commutativity: a \u22c5b = b \u22c5a \u2200a, b \u2208G hold in G<\/p>\n\n\n\n

1 \u22c51 = 1 = 1 \u22c51, \u22121 \u22c51 = \u22121 = 1 \u22c5\u22121<\/p>\n\n\n\n

i\u22c51 = i= 1 \u22c5i; i\u22c5\u2212i= \u2212i\u22c5i= 1 = 1 etc.<\/p>\n\n\n\n

Commutative law is satisfied.<\/p>\n\n\n\n

Hence (G, \u22c5) is an abelian group.<\/p>\n\n\n\n

Example<\/strong><\/h2>\n\n\n\n

Prove that the set Z of all integers with binary operation * defined by a * b = a + b + 1 <\/strong>\u2200<\/strong>a, b <\/strong>\u2208<\/strong>G is an abelian group.<\/strong><\/p>\n\n\n\n

Sol: <\/strong>Sum of two integers is again an integer; therefore a +b \u2208Z \u2200a, b \u2208Z<\/p>\n\n\n\n

\u21d2a +b + 1 \u22c5\u2208Z \u2200a, b \u2208Z<\/p>\n\n\n\n

\u21d2Z is called with respect to *<\/p>\n\n\n\n

Associative law for all a, b, a, b https:\/\/upinsmoke.net.au\/<\/a> \u2208G we have (a * b) * c = a * (b * c) as<\/p>\n\n\n\n

(a* b) * c = (a + b + 1) * c<\/p>\n\n\n\n

= a + b + 1 + c + 1<\/p>\n\n\n\n

= a + b + c + 2<\/p>\n\n\n\n

Also<\/p>\n\n\n\n

a* (b * c) = a * (b + c + 1)<\/p>\n\n\n\n

= a + b + c + 1 + 1<\/p>\n\n\n\n

= a + b + c + 2<\/p>\n\n\n\n

Hence (a * b) * c = a * (b * c) \u2208a, b \u2208Z.<\/p>\n\n\n\n

Subgroup<\/strong><\/h2>\n\n\n\n

Let (G, *) be a group and H, be a non-empty subset of G. If (H, *) is itself is a group, then (H, *) is called sub-group of (G, *).<\/p>\n\n\n\n

Example-Let a = {1, \u20131, i, \u2013i} and H = {1, \u20131}<\/strong><\/p>\n\n\n\n

G and H are groups with respect to the binary operation, multiplication.<\/strong><\/p>\n\n\n\n

H is a subset of G, therefore (H, X) is a sub-group (G, X).<\/strong><\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n