# How do you prove it is a subgroup?

Table of Contents

## How do you prove it is a subgroup?

Subgroup tests Suppose that G is a group, and H is a subset of G. Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. (Closed under products means that for every a and b in H, the product ab is in H. Closed under inverses means that for every a in H, the inverse a−1 is in H.

## What is the order of a group G?

The Order of a group (G) is the number of elements present in that group, i.e it’s cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the identity element of the group, and an denotes the product of n copies of a.

**How many subgroups does a group of order 11 have?**

Prove that G has at least 11 elements of order 11. So by First Sylow’s theorem, there exists a Sylow 11-subgroup of G. By Third Sylow’s theorem, the number of such subgroups is 11k+1 and 11k+1|1331, thus, this is only possible for k=0.

### What is the order of a subgroup?

In general, the order of any subgroup of G divides the order of G. More precisely: if H is a subgroup of G, then ord(G) / ord(H) = [G : H], where [G : H] is the index of H in G, an integer. This is Lagrange’s theorem. If a has infinite order, then all powers of a have infinite order as well.

### Is HK a normal subgroup?

Thus gxg−1 = g(hk)g−1 = (ghg−1)(gkg−1) ∈ HK (since ghg−1 ∈ H and gkg−1 ∈ K). This proves that HK is a normal subgroup.

**What is subgroup order?**

In general, the order of any subgroup of G divides the order of G. More precisely: if H is a subgroup of G, then. ord(G) / ord(H) = [G : H], where [G : H] is called the index of H in G, an integer. This is Lagrange’s theorem. (This is, however, only true when G has finite order.

#### How many subgroups are there in a group of order 13?

one subgroup

We know that there is only one subgroup of order 13(By Sylow’s thm) which implies there are exactly 12 elements of order 13 (precisely the non-identity elements of the subgroup of order 13). Now every element has either order=3 or order=13 or order=1 (by Lagrange’s thm).

#### How many subgroups are there for a group of order 7?

The 8 Sylow subgroups of order 7 each contain 6 elements of order 7 together with the identity. Since an element of order 7 generates a group of order 7 these elements are all distinct.

**Is union of two subgroups A subgroup?**

Union of two subgroups is not a subgroup unless they are comparable – Groupprops.