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Chapter 9: Q9.2-3E-c (page 297)

List all abelian groups (up to isomorphism) of the given order:30

Short Answer

Expert verified

The abelian group of order 30 is $ℤ_{2} \oplus ℤ_{3} \oplus ℤ_{5}$ .

Step by step solution

01

Step-by-Step Solution Step 1: Finite Abelian Group

Every finite abelian group G is the direct sum of cyclic groups, each of prime power orders.

02

Abelian groups of order 30

The number 30 can be written as a product of prime powers as, $30 = 2 \cdot 3 \cdot 5$ .

Hence, by the above theorem, the abelian group can be written as: $ℤ_{2} \oplus ℤ_{3} \oplus ℤ_{5}$ .

Therefore, the abelian group of order 30 is $ℤ_{2} \oplus ℤ_{3} \oplus ℤ_{5}$ .

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Most popular questions from this chapter

Let $G$ be a group and $f_{1} : G \to G_{1}, f_{2} : G \to G_{2}, \dots, f_{n} : G \to G_{n}$ homomorphisms. For $i = 1, 2, \dots, n$ , let $π_{i}$ be the homomorphism of Exercise 8. Let $f^{*} : G \to G_{1} \times \dots \times G_{n}$ be the map defined by $f^{*} (a) = (f_{1} (a_{1}), f_{2} (a_{2}), \dots, f_{n} (a_{n}))$ .

Prove that $f^{*}$ is the unique homomorphism from $G$ to $G_{1} \times \dots \times G_{n}$ such that $π_{i} \circ f^{*} = f_{i}$ for every $i$ .

Prove that there are no simple groups of the given order:200

Find the order of each element in the given group:

(c) $D_{4} \times ℤ_{2}$

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.

Let $i$ be an integer with $1 \leq i \leq n$ . Let ${\bar{G}}_{i}$ be the subset of $G_{1} \times G_{2} \times \dots \times G_{n}$ consisting of those elements whose $i$ th coordinate is any element $G_{i}$ of and whose other coordinates are each the identity element, that is,

${\bar{G}}_{i} = {(e_{1}, e_{2}, \dots, e_{i - 1}, e_{i}, e_{i + 1}, \dots, e_{n}) | e_{i} \in G_{i}}$

Prove that,

${\bar{G}}_{i} ≅ G_{i}$

See all solutions

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