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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.
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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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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.
How do we prove that a finite group G of order p prime is cyclic using Lagrange?
Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of G with more then 1 element is G itself, therefore G is cyclic. QED.
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.
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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?
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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
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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?
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What family does the element titanium belong to?
Titanium belongs to the family of transition metals.
