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tehmeenasidiq

Answered question

2022-05-26

How many non cyclic groups of order 14 are

Answer & Explanation

Eliza Beth13

Skilled2023-05-17Added 130 answers

To determine the number of non-cyclic groups of order 14, we can use the concept of group isomorphism and the fundamental theorem of finite abelian groups.
Let's start by prime factorizing the order of the group, which is 14:

14 = 2 \times 7

According to the fundamental theorem of finite abelian groups, any finite abelian group can be decomposed into a direct product of cyclic groups. For an abelian group of order

p_{1}^{k_{1}} \times p_{2}^{k_{2}} \times \dots \times p_{n}^{k_{n}}

, where

p_{i}

are distinct prime numbers, the number of possible non-isomorphic groups is given by:

Number of non-isomorphic groups = (k_{1} + 1) (k_{2} + 1) \dots (k_{n} + 1)

In our case, we have

14 = 2 \times 7

. Therefore, the number of non-isomorphic groups of order 14 is:

Number of non-isomorphic groups = (1 + 1) (1 + 1) = 2 \times 2 = 4

Hence, there are 4 non-cyclic groups of order 14.

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