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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
- UGC NET
- Computer Science & Applications
What is group
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A group is a finite or infinite set of elements together with a binary operation (called the group operation) on that set that together satisfy the fundamental properties. 1. Closure 2. Associativity 3. Identity property and 4. Inverse property. Suppose, G be a non-void set with a binary operation * that assigns to each ordered pair (a, b) of elements of G an element of G denoted by a * b. We say that G is a group under the binary operation * if the following three properties are satisfied: 1) Associativity :- The binary operation * is associative i.e. a*(b*c)=(a*b)*c , ∀ a,b,c ∈ G 2) Identity :- There is an element e, called the identity, in G, such that a*e=e*a=a, ∀ a ∈ G 3) Inverse :- For each element a in G, there is an element b in G, called an inverse of a such that a*b=b*a=e, ∀ a, b ∈ G
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sub group means
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Subgroup :- Let H be a nonempty subset of a group G. If H also becomes a group under the same binary operation (*) as in G, then H is called as a subgroup of G. We use the notation H < =G to indicate that H is a subgroup of G. A subgroup H is a subset of a group G (denoted by H < =G ) if it satisfies the four properties simultaneously -Closure, Associative, Identity, Inverse