In which of the following sets, every element has multiplicative inverse?
(a) ℤ (b) ℚ (c). ℝ (d). ℝ = ℝ\{0}
Answers
Answer:
A group consists of a non-empty set G along with a binary operation "* " ( * : G� G ® G), satisfying the following group axioms:
(1) If a, b, c Î G, then a* (b* c) = (a* b)* c; i.e., * is associative.(2) There exists an element e Î G such that for any a Î G, a* e = e* a = a.(3) For any a Î G, there exists an element a Î G such that a* a = a * a = e.
It is easy to verify that e, called the identity element of the group G, is unique. For any a Î G, the element a as in (3), called an inverse of a, is also unique.
A group is called finite if it has only a finite number of elements in it. Otherwise, we say that the group is infinite. The number of elements in a group, or its cardinality, is called the order of the group.
The sets of complex numbers C, real numbers R, rational numbers Q and integers Z are groups under the binary operation "+", the ordinary addition of real numbers. In this case e = 0 and a = -a. The sets of non-zero real numbers, non-zero rational numbers, positive real numbers, positive rational numbers etc. are groups under the binary operation "� ", the ordinary multiplication of real numbers. In this case e = 1 and a = a-1 = 1/a. These two examples, utilizing the ordinary addition and multiplication, have motivated the phrases: additive group (the binary operation "* " is written as "+", e is called the zero or the additive identity and is written as 0, a is called the negative or the additive inverse of a and is written as -a) and multiplicative group (in which case, the binary operation * is usually written as ". " or is omitted altogether (i.e., a.b or ab represent a* b), e is called the unity or the multiplicative identity and is written as 1; a is called the multiplicative inverse of a and is written as a-1). In a multiplicative group am means a* a* ... * a (m-times). Thus a3 = a* a* a. In an additive group, similarly, ma denotes the addition or sum of m number of a's.
A group G is called commutative if for all a, b Î G there holds a* b = b* a, i.e., a and b commute.
A subgroup H of a group G is a subset which under the same binary operation is itself a group. Two groups G and G' are said to be isomorphic if there exists a 1-1 onto map h : G ® G' such that h (ab) = h (a)h (b) for all a, bÎ G. We say that h is an isomorphism from G to G', while it follows that h -1 is an isomorphism from G' to G.
Step-by-step explanation: