Chapter 7: Q 14b E (page 211)

(b) Let H and K be subgroups of a group G. Prove that $H \cup K$ is a subgroup of G if and only if $H \subseteq K or K \subseteq H$ .

Short Answer

Expert verified

We prove that $H \cup K$ is a subgroup of G if and only if $H \subseteq K o r K \subseteq H$ .

Step by step solution

To show eitherH⊆K or K⊆H

Let $H \cup K$ be a subgroup of group G.

To show, either $H \subseteq K o r K \subseteq H$ .

Suppose that $H \emptyset K and K \emptyset H$ .

Now, $H \emptyset K \Rightarrow \exists a \in H$ such that $a \notin K$ and

$K \emptyset H \Rightarrow \exists b \in K$ such that $b \notin H$ .

$a \in H \Rightarrow a \in H \cup K$ and $b \in K \Rightarrow b \in H \cup K$

Thus, a,b $\in H \cup K$ and $H \cup K$ is a subgroup.

$\Rightarrow a * b^{- 1} \in H \cup K \Rightarrow a * b^{- 1} \in H or a * b^{- 1} \in K$

Case 1:

$a * b^{- 1} \in H a^{- 1} * (a * b^{- 1}) \in H . . . (since, H is a subgroup and a \in H) (a^{- 1} * a) * b^{- 1} \in H e * b^{- 1} \in H b^{- 1} \in H {(b^{- 1})}^{- 1} \in H . . . (since, H is a subgroup) b \in H Therefore, a contradiction, since b \notin H$

Case 2:

$a * b^{- 1} \in K (a * b^{- 1}) * b \in K . . . (since, K is a subgroup and b \in K) a * (b^{- 1} * b) \in K a * e \in K a \in K Therefore, a contradiction, since, a \notin K$

Thus, in any case, we get a contradiction.

Hence, either $H \subseteq K or K \subseteq H$

To show that H∪Kis a subgroup of G

Conversely, suppose that either $H \subseteq K o r K \subseteq H$ .

Therefore, either $H \cup K = K or H \cup K = H$

Therefore, $H \cup K$ is a subgroup of G.

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(b) Let H and K be subgroups of a group G. Prove that $H \cup K$ is a subgroup of G if and only if $H \subseteq K or K \subseteq H$ .

Short Answer

Step by step solution

To show eitherH⊆K or K⊆H

To show that H∪Kis a subgroup of G

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(b) Let H and K be subgroups of a group G. Prove that H∪Kis a subgroup of G if and only if H⊆KorK⊆H.

Short Answer

Step by step solution

To show eitherH⊆K or K⊆H

To show that H∪Kis a subgroup of G

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Statistics

Geometry

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

(b) Let H and K be subgroups of a group G. Prove that $H \cup K$ is a subgroup of G if and only if $H \subseteq K or K \subseteq H$ .