Table of Contents

- 1 How many subgroups are in a group?
- 2 How many subgroups are there for a group of order 19?
- 3 What is subgroup order?
- 4 How many Sylow 3 − subgroups are there in a Noncyclic group of order 21?
- 5 What are the 8 groups of the periodic table?
- 6 How many subgroups does Order 2 have?
- 7 How many distinct subgroups are there in a cyclic group?
- 8 What are the orders of the elements of a symmetric group?

## How many subgroups are in a group?

In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup’s order is a divisor of n, and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups.

**How do I find the number of subgroups in 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 subgroups are there for a group of order 19?

Lagrange’s theorem states that the order of any subgroup of a group is a factor of the order of the group. Since 19 is prime it has only two factors 1 and 19. Therefore the group can have only 2 subgroups.

**What is the group number of M?**

Fact box

Group | 7 | 1246°C, 2275°F, 1519 K |
---|---|---|

Period | 4 | 2061°C, 3742°F, 2334 K |

Block | d | 7.3 |

Atomic number | 25 | 54.938 |

State at 20°C | Solid | 55Mn |

#### What is subgroup order?

Lagrange’s theorem states that for any subgroup H of a finite group G, the order of the subgroup divides the order of the group; that is, |H| is a divisor of |G|. In particular, the order |a| of any element is a divisor of |G|.

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

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 Sylow 3 − subgroups are there in a Noncyclic group of order 21?

Solution: The number of Sylow 3-subgroups is equal to 1 mod 3 and divides 7. Thus there are either 1 or 7 such subgroups.

**How many groups are there of order 15?**

Table of number of distinct groups of order n

Order n | Prime factorization of n | Number of groups |
---|---|---|

13 | 13 1 | 1 |

14 | 2 1 ⋅ 7 1 | 2 |

15 | 3 1 ⋅ 5 1 | 1 |

16 | 2 4 | 14 |

### What are the 8 groups of the periodic table?

The following are the 8 groups of the periodic table:

- Alkali metals.
- Alkaline earth metals.
- Rare earth metals.
- Crystallogens.
- Pnictogens.
- Chalcogens.
- Halogens.
- Noble gases.

**Why are there 18 groups in the periodic table?**

The same was found of beryllium, magnesium and calcium. Fluorine, chlorine and bromine. Oxygen and sulfur. Look at your modern periodic table and notice that these groups of elements all happen to share the same column.

#### How many subgroups does Order 2 have?

(5) There are 5 groups of order 2, because there are 4 elements of order 2. These are the subgroups generated by x, y, a, d, and r2.

**What is the Order of the subgroups of a group?**

In this case, the order of a group is a prime number. That makes the problem very easy. The only divisors of a prime number are 1 and the prime number itself. Therefore, every group with a prime order has exactly two subgroups, namely, the group itself and the trivial subgroup that has as its only element the identity of the group.

## How many distinct subgroups are there in a cyclic group?

Thus, there are 1 (identity subgroup) + 1 (the group itself) + 1 (subgroup of order 2) + 1 (subgroup of order 3) = 4 distinct subgroups in total. Let Zn be a cyclic group of order n. For each divisor of n there exists unique subgroup of Zn.

**How do you find the Order of a group of elements?**

Order of element a ∈ G is the smallest positive integer n, such that a n = e, where e denotes the identity element of the group, and a n denotes the product of n copies of a. If no such n exists, a is said to have infinite order. All elements of finite groups have finite order.

### What are the orders of the elements of a symmetric group?

For example, in the symmetric group shown above, where ord(S3) = 6, the orders of the elements are 1, 2, or 3. The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy’s theorem).