Total number of non abelian groups of order 23 . 34. 5 is __________.Correct answer is '15'. Can you explain this answer?

Question

Answer

Partitions of 3 are three namely

3 = 3,2 + 1, 1 +1 +1.

Partition of 4 are five namely

4 = 4, 3 + 1, 2 + 2, 2 + 1 +1, 1 +1 +1 +1.

Partition of 1 is only one 1 = 1

Thus there are (I) non isomorphic abelian group of order 2.

(II) five non isomorphic abelian group of order 3⁴. and

(III) One abelian group of order 5.

Hence total number of non abelian group of order

2³. 3⁴. 5 = 3 x 5 x 1 = 1

=15

Answer

Partitions of 3 are three namely

3 = 3,2 + 1, 1 +1 +1.

Partition of 4 are five namely

4 = 4, 3 + 1, 2 + 2, 2 + 1 +1, 1 +1 +1 +1.

Partition of 1 is only one 1 = 1

Thus there are (I) non isomorphic abelian group of order 2.

(II) five non isomorphic abelian group of order 3⁴. and

(III) One abelian group of order 5.

Hence total number of non abelian group of order

2³. 3⁴. 5 = 3 x 5 x 1 = 1

=15

Answer

The total number of non-abelian groups of order 23 * 34 * 5 is 15. Let's break down the steps to understand why this is the correct answer.

Step 1: Prime Factorization
First, we need to find the prime factorization of the given number. The prime factorization of 23 * 34 * 5 is as follows:

23 = 23^1
34 = 2^1 * 17^1
5 = 5^1

Step 2: Counting Non-Abelian Groups
To determine the number of non-abelian groups of a given order, we need to consider the possible combinations of prime powers.

For a prime power p^n, there are two possibilities:
1. The group is abelian, which means it is isomorphic to a direct product of cyclic groups of prime power order.
2. The group is non-abelian.

Since we are interested in counting only the non-abelian groups, we need to exclude the abelian ones.

Step 3: Counting Abelian Groups
To count the number of abelian groups, we need to consider each prime power separately.

For the prime power 23^1, there is only one abelian group of order 23^1, which is the cyclic group of order 23.

For the prime power 2^1, there are two abelian groups of order 2^1: the cyclic group of order 2, and the trivial group.

For the prime power 17^1, there is only one abelian group of order 17^1, which is the cyclic group of order 17.

For the prime power 5^1, there is only one abelian group of order 5^1, which is the cyclic group of order 5.

Step 4: Calculating Non-Abelian Groups
To calculate the number of non-abelian groups, we subtract the number of abelian groups from the total number of groups.

For the prime power 23^1, there is only one group of order 23^1.

For the prime power 2^1, there is only one group of order 2^1.

For the prime power 17^1, there is only one group of order 17^1.

For the prime power 5^1, there is only one group of order 5^1.

Multiplying these counts together, we get 1 * 1 * 1 * 1 = 1.

Therefore, the total number of non-abelian groups of order 23 * 34 * 5 is 1.

However, it is mentioned that the correct answer is 15, which means there may be a mistake in the given information or question. Please double-check the question or provide more context if necessary.

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