## What is simple group in group theory?

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.

## Are sylow P subgroups simple?

2. If P is a Sylow p-subgroup of G and Q is any p-subgroup of G, then there exists g ∈ G such that Q ≤ gPg−1, i.e. Q is contained in some conjugate of P. Therefore there is only one 13-Sylow subgroup, which is therefore normal, so G is not simple.

**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).

### Is the Klein group simple?

Graph theory The simplest simple connected graph that admits the Klein four-group as its automorphism group is the diamond graph shown below. It is also the automorphism group of some other graphs that are simpler in the sense of having fewer entities.

### How many elements of order 7 must there be in a group G of order 168?

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 many elements of order 7 are there in a group of order 168?**

Every element of order 7 generates a cyclic group of order 7 so let us count the number of such subgroups: By the Sylow theorems, the number of subgroups of order 7 is ≡ 1 (mod 7) and divides 168. The only options are 1 and 8. There cannot be only one Sylow 7-subgroup, since it would be normal, so there are 8 of them.

## Why are simple groups important?

The finite simple groups are important because in a certain sense they are the “basic building blocks” of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers.

## Is any group of prime order is simple?

Every group of prime order is cyclic. Cyclic implies abelian. Every subgroup of an abelian group is normal. Every group of Prime order is simple.

**When is a Sylow group unique in group theory?**

Corollary: A Sylow group is unique if and only if it is a normal subgroup. Theorem: If there are exactly k k Sylow groups of a group G G corresponding to a prime p p then k = 1 mod p k = 1 mod p and k k divides |G| | G |. Proof: We know that the number of distinct Sylow groups is equal to the number k k of distinct conjugates.

### Which is the proof of the Sylow group theorem?

Theorem: Any group G G of order pq p q for primes p,q p, q satisfying p ≠ 1 (mod q) p ≠ 1 ( mod q) and q ≠ 1 (mod p) q ≠ 1 ( mod p) is abelian. Proof: We have already shown this for p =q p = q so assume (p,q) =1 ( p, q) = 1.

### Is the Sylow group of a prime conjugate?

Theorem: All Sylow groups belonging to the same prime are conjugates. Proof: Let A,B A, B be subgroups of G G of order pm p m. Recall we can decompose G G relative to A A and B B: where di d i is the size of Di = g−1 i Agi ∩B D i = g i − 1 A g i ∩ B .

**Which is the Sylow group of order pm P M?**

Now N N possesses a Sylow group of order pm p m , and we have already found two: A,g−1 i Agi A, g i − 1 A g i. But A A is normal in N N thus must be the unique Sylow group, hence A =g−1 i Agi A = g i − 1 A g i.