How many elements of order 7 can a group of order 168 have?

Since an element of order 7 generates a group of order 7 these elements are all distinct. There are 8×6=48 elements of order 7.

How do you find the subgroup of G?

A subgroup H of the group G is a normal subgroup if g -1 H g = H for all g ∈ G. If H < K and K < G, then H < G (subgroup transitivity). if H and K are subgroups of a group G then H ∩ K is also a subgroup.

How do you prove a group is simple?

A group G is simple if its only normal subgroups are G and 〈e〉. A Sylow p-subgroup is normal in G if and only if it is the unique Sylow p-subgroup (that is, if np = 1).

How do you find the number of subgroups of a group?

In order to determine the number of subgroups of a given order in an abelian group, one needs to know more than the order of the group, since for example there are two different groups of order 4, and one of them has one subgroup of order 2, which the other has 3.

How many elements of order 7 are there in a group of Order 28?

Given a group G with order 28=22⋅7. Sylow-Theory implies that there is a exactly one 7-Sylow-Subgroup of order 7 in G, and 1 or 7; 2-Sylow-Subgroups.

What does it mean for a group to be normal?

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all. and.

What is the easiest way to find subgroups?

The most basic way to figure out subgroups is to take a subset of the elements, and then find all products of powers of those elements. So, say you have two elements a,b in your group, then you need to consider all strings of a,b, yielding 1,a,b,a2,ab,ba,b2,a3,aba,ba2,a2b,ab2,bab,b3,…

What does it mean if a group is simple?

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group.

Is every group of prime order is simple?

Every subgroup of an abelian group is normal. Every group of Prime order is simple.

How many subgroups does G have?

Note: Every group G has at least two subgroups: G itself and the subgroup {e}, containing only the identity element.

How do you find the number of elements in a given order?

If n divides the order of a group, then the number of elements in the group whose orders divide n is a multiple of n. We call G a minimal counterexample. We proceed to contradict the minimality of G, and thus conclude that such G, in fact, does not exist. then n divides the difference Nnp-Nn’=Nn.