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It is a subset of the Group G which has all the properties of a Group, namely that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element which is also in the set. The Group satisfies four axioms: closure, associativity, identity and invertibility.

The set of integers, Z, is a Group, with addition as the binary operation. [It is also a Ring, but that is not important here]. The set of all multiples of 7 is a subgroup of Z.

Denote the subgroup by Z7. It is a Group because:

Closure: If x and y are in Z7, then x = 7*p for some p in Z and y = 7*q for some q in Z. Then x + y = 7*p + 7*q = 7*(p+q) where p+q is in Z because Z is a Group. Therefore 7*(p+q) is in Z7.

Associativity: If x (= 7p), y (= 7q) and z (= 7r) are in 7Z, then

(x + y) + z = (7p + 7q) + 7r

since these are in Z an Z is associative, = 7p + (7q + 7r) = x + (y + z).

Identity: The additive identity is 0, since 0 + x = 0 + 7p = 7p since 0 is the additive identity in Z.

Invertibility: If x = 7a is in Z7 then 7*(-a) is also in 7Z. If 7*(-a) is denoted by -x, then x + (-x) = 7a + 7*(-a) = 0 and so -x is the additive inverse of x.

But there are elements of Z, for example, 2 which are not in Z7 so Z7 it is a proper subset of Z.

Q: What is meant by a subgroup generated by an element x belongs to group G?

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The properties of a subgroup would include the identity of the subgroup being the identity of the group and the inverse of an element of the subgroup would be the same in the group. The intersection of two subgroups would be a separate group in the system.

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The properties of a subgroup would include the identity of the subgroup being the identity of the group and the inverse of an element of the subgroup would be the same in the group. The intersection of two subgroups would be a separate group in the system.

In abstract algebra, a generating set of a group Gis a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.More generally, if S is a subset of a group G, then , the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.If G = , then we say S generatesG; and the elements in S are called generators or group generators. If S is the empty set, then is the trivial group {e}, since we consider the empty product to be the identity.When there is only a single element x in S, is usually written as . In this case, is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that it has order |G|, or that equals the entire group G.My source is linked below.

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