Is the group of order 42 Simple?

For groups of order 40, 42 and 44, the argument is the following. Say |G| = 42 = 6 · 7, then s7 divides 6 and s7 ≡ 1 mod 7. Hence s7 = 1, which is equivalent to say that that G has a normal Sylow 7-subgroup, hence it is not a simple group.

What is the order of a cyclic group?

Definition and notation The order of g is the number of elements in ⟨g⟩; that is, the order of an element is equal to the order of its cyclic subgroup. A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator.

What is the cyclic group of order 2?

The cyclic group of order 2 may occur as a normal subgroup in some groups. Examples are the general linear group or special linear group over a field whose characteristic is not 2. This is the group comprising the identity and negative identity matrix. It is also true that a normal subgroup of order two is central.

Does a group of order 14 have to be cyclic?

Any abelian group of order 14 is cyclic. Any abelian group of order 21 is cyclic.

How many elements of order 5 does S7 have?

How many permutations of order 5 are there in S7? = 21.

How many Abelian groups up to isomorphism are there of order 6?

11.38 Note that up to isomorphism, there are 6 Abelian groups of order 72, namely, G1 × G2 for G1 ∈ {ZZ8,ZZ2 × ZZ4,ZZ2 × ZZ2 × ZZ2}, and G2 ∈ {ZZ9,ZZ3 × ZZ3}. (a) Suppose H ≤ G1 × G2 has order 8.

Do all cyclic groups have prime order?

The statement you claim to have contradicted, i.e. that every element of a cyclic group G has order either 1 or |G|, is false.

Is z * z cyclic?

Since the dim(ZxZ)=2>dim(Z)=1, we know that ∄ an isomorphism between our spaces. Hence, ZxZ is not a cyclic group.

How many elements of order 2 are in a cyclic group?

(1) A cyclic group cannot have more than one element of order 2. The number of elements in a cyclic group of order 2 can be only ϕ(2) =1.

Is Z * Z cyclic?

How many elements of order 5 are there in a?

So really, there are only 5! 5 distinct 5-cycles on a given set of elements. Therefore, there are 7!

Which is the best definition of a cyclically ordered group?

Cyclically ordered groups. A cyclically ordered group is a group together with a cyclic order preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group).

Is the field extension of the rational numbers cyclic?

The Galois group of the field extension of the rational numbers generated by the n th roots of unity forms a different group, isomorphic to the multiplicative group (Z/nZ) × of order φ(n), which is cyclic for some but not all n (see above). A field extension is called a cyclic extension if its Galois group is cyclic.

What are the properties of a cyclic group?

Properties of Cyclic Groups Definition (Cyclic Group). A group G is called cyclic if 9 a 2 G 3 G = hai = {an|n 2 Z}. We say a is a generator of G. (A cyclic group may have many generators.) Although the list …,a 2,a 1,a0,a1,a2,… has infinitely many entries, the set {an|n 2 Z} may have only finitely many elements.

Is the group operation of a cyclic group commutative?

Every cyclic group is abelian. That is, its group operation is commutative: gh = hg (for all g and h in G). This is clear for the groups of integer and modular addition since r + s ≡ s + r (mod n), and it follows for all cyclic groups since they are all isomorphic to these standard groups.