Question

Theorems 8.7, 9.7, 9.30 and 9.33, and Corollaries 9.18 and 9.29 are sufficient to classify groups of many orders. List all such orders from 16 to 100.

Short answer

We listed all the 44 orders of groups from 16 to 100.

Step-by-step solution

Recalling all the required Theorems and Corollaries

Definition of Subgroup

Let be a group under binary operation, say . A non-empty subset of is said to be a subgroup of , if is itself a group.

In other words, if and , then .

Definition of Abelian Group

A group is called abelian if , .

Theorem 8.7states that, let be a positive prime integer. Every group of order is cyclic and isomorphic to .

Theorem 9.7 states that, every finite abelian group is the direct sum of cyclic groups, each of prime power order.

Corollary 9.18, let be a group of order , where and are primes such that . If , then .

Corollary 9.29 gives, if is a group of order , with prime , is isomorphic to or .

Theorem 9.30 states that, let and be distinct primes such that and . If is a group of order , then is isomorphic to or .

Theorem 9.33, if is a group of order , where is an odd prime, then is isomorphic to the cyclic group or the dihedral group .

To list out all the orders of groups from 16 to 100

The theorem and corollaries mentioned in step 1 helps to classify groups of orders from 16 to 100, which are listed as follows:

$$\begin{gathered} 17=17 \\ 19=19 \\ 22=2.11 \\ 23=23 \end{gathered}$$

Further,

$$\begin{gathered} 25=5^2 \\ 26=2.13 \\ 29=29 \\ 31=31 \end{gathered}$$

Remaining groups are written as:

$$\begin{gathered} 33=3.11 \\ 34=2.17 \\ 35=5.7 \\ 37=37 \end{gathered}$$

Further,

localid="1653050478657" $$\begin{gathered} 38=2.19 \\ 41=41 \\ 43=43 \\ 45=3^2.5 \end{gathered}$$

Remaining groups are written as:

$$\begin{gathered} 46=2.23 \\ 47=47 \\ 49=7^2 \\ 51=3.17 \end{gathered}$$

Further,

$$\begin{gathered} 53=53 \\ 58=2.29 \\ 59=59 \\ 61=61 \end{gathered}$$

Remaining groups are written as:

$$\begin{gathered} 62=2.31 \\ 65=5.13 \\ 67=67 \\ 69=3.23 \end{gathered}$$

Further,

$$\begin{gathered} 71=71 \\ 73=73 \\ 74=2.37 \\ 77=7.11 \end{gathered}$$

Remaining groups are written as:

$$\begin{gathered} 79=79 \\ 82=2.41 \\ 83=83 \\ 85=5.17 \end{gathered}$$

Further,

$$\begin{gathered} 86=2.43 \\ 87=3.29 \\ 89=89 \\ 91=7.13 \end{gathered}$$

Remaining groups are written as:

$$\begin{gathered} 94=2.47 \\ 95=5.19 \\ 97=97 \\ 99=3^2.11 \end{gathered}$$

Hence, we listed all the 44 orders of groups from 16 to 100.