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Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
Is every abelian group is cyclic or not and why?
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
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.
Is every abelian group is cyclic or not?
No.
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.
Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
Is every abelian group is cyclic or not and why?
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
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.
Is every abelian group is cyclic or not?
No.
Prove that every a cyclic group is an abelia?
Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a
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.
Is every finite abelian group is cyclic?
No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic
What is Cayley's Theorem?
Cayley's Theorem states that every group G is isomorphic to a subgroup of the symmetric group on G.
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.
Prove that a group of order 5 must be cyclic?
There's a theorem to the effect that every group of prime order is cyclic. Since 5 is prime, the assertion in the question follows from the said theorem.
What is the subgroup for class?
The term "subgroup" typically refers to a smaller group within a larger group. In the context of "class," a subgroup could refer to a smaller group of students within a class who are working on a specific project or assignment together.
A cyclic group of length 2 is called identity?
A cyclic group of order two looks like this.It has two elements e and x such that ex = xe = x and e2 = x2 = e.So it is clear how it relates to the identity.In a cyclic group of order 2, every element is its own inverse.
