# Is the group of order 42 Simple?

## 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.