0
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.
What else can I help you with?
What are the properties of a subgroup?
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 group theory what is a group generator?
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.
Every subgroup of a cyclic group is cyclic?
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
What is subgroup in mathematics?
In mathematics, a subgroup H of a group G is a subset of G which is also a group with respect to the same group operation * defined on G. H contains the identity element of G, is closed with respect to *, and all elements of H have their inverses in H as well.
Can a non-abelian group have a torsion subgroup?
Yes, a non-abelian group can have a torsion subgroup. A torsion subgroup is defined as the set of elements in a group that have finite order. Many non-abelian groups, such as the symmetric group ( S_3 ), contain elements of finite order, thus forming a torsion subgroup. Therefore, the existence of a torsion subgroup is not restricted to abelian groups.
What are the properties of a subgroup?
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.
What group of dinosaurs was arcyopteryex?
Archeopteryx belongs to the group Aves, which are a subgroup of Theropoda. They were closely related to the link between dinosaurs and birds.
In group theory what is a group generator?
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.
Prove that the order of an element of a group of finite order is a divisor of the order of the group?
Let ( G ) be a finite group with order ( |G| ), and let ( g \in G ) be an element of finite order ( n ). The order of ( g ), denoted ( |g| ), is the smallest positive integer such that ( g^k = e ) for some integer ( k ), where ( e ) is the identity element. The subgroup generated by ( g ), denoted ( \langle g \rangle ), has order ( |g| = n ). By Lagrange's theorem, the order of any subgroup divides the order of the group, thus ( |g| ) divides ( |G| ).
What group does platinum belong in?
this element belongs in the group of metals
Every subgroup of a cyclic group is cyclic?
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
Element Nitrogen belongs to what family?
Nitrogen belongs to Group 5 or Pnictogens
What element has 3 energy levels and belongs to the most unreactive group?
The element is 'argon' and belongs to noble gas family.
What criteria did you use to choose group an element belongs to?
²⅔
What group of organic compounds does cholesterol belong to?
Cholesterol belongs to the group of organic compounds known as sterols. Sterols are a subgroup of steroids that contain a hydroxyl group at the C-3 position of the A-ring.
What is subgroup in mathematics?
In mathematics, a subgroup H of a group G is a subset of G which is also a group with respect to the same group operation * defined on G. H contains the identity element of G, is closed with respect to *, and all elements of H have their inverses in H as well.
What group does platinum belong to?
this element belongs in the gruop of metals
