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Chapter 8: Q44E-b (page 247)

Prove that the elements of order 2 and the identity element form a subgroup.

Short Answer

Expert verified

It is proved that, these three elements along with the identity from a subgroup of $A_{4}$ .

Step by step solution

Obtain the elements of order 2 and the identity element form a subgroup

It follows the part (a) and exercise 12 part (a).

Hence, It is proved that these three elements along with the identity from a subgroup of $A_{4}$ .

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Most popular questions from this chapter

Let $G$ be an abelian group of order $n$ and let $K$ be a positive integer. If $(k, n) = 1$ , prove that the functionrole="math" localid="1654351034332" $f : G \to G$ given by $f (a) = a^{k}$ is an isomorphism.

Question:In Exercise 13-15, K is a subgroup of G . Determine whether the given cosets are disjoint or identical.

14. $G = S_{4}$ ; K is the subgroup ${(), (12) (34), (13) (24), (14) (23)}$

(b) $K (1234)$ and $K (1324)$ .

Let $k$ b e a subgroup of a group $G$ and let $a \in G$ . Prove thatrole="math" localid="1654334784177" $k a = k$ if and only if $a \in k$ .

State the number of co sets of $k$ in $A_{4}$ . Don't list them.

In Exercises 7-11 $G$ is a group and $H$ is a subgroup of G. Find the index $[G : H]$ .

9. $H = ⟨ 3 ⟩; G = ℤ_{20}$

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Prove that the elements of order 2 and the identity element form a subgroup.

Short Answer

Step by step solution

Obtain the elements of order 2 and the identity element form a subgroup

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Most popular questions from this chapter

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