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Chapter 9: Q14E (page 319)

Show that every subgroup of the quaternion group Q is normal.

Short Answer

Expert verified

It is proved that, every subgroup of the quaternion group Q is normal.

Step by step solution

01

Prove every subgroup of the quaternion group Q is normal

A quaternion group is a non abelian group of order eight and it is isomorphic to {1,-1,i,-i,j,-j,k,-k}.

Clearly, the subgroups $<e>$ and Q are normal.

Any subgroup of order 4 is also normal.

It can be seen that, -1 is the only element of order 2.

Therefore, there is only one subgroup of order 2 in Q, which is definitely normal.

Since all the possibilities are covered, it can be concluded that all the subgroups of Q are normal.

Hence, every subgroup of the quaternion group Q is normal.

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Most popular questions from this chapter

Question: If $a \in G$ , then show by example that may not be abelian. [Hint: If role="math" localid="1653318623161" $a = (12)$ in role="math" localid="1653318640031" $S_{5}$ , then role="math" localid="1653318666049" $(34)$ and role="math" localid="1653318676522" $(345)$ are in role="math" localid="1653318690379" $C (a)$ .]

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 of $G_{i}$ 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

${\bar{G}}_{i}$ is a normal subgroup of $G_{1} \times \dots \times G_{n}$ .

Find the order of each element in the given group:

(a) $ℤ_{2} \times ℤ_{4}$

In the proof of Theorem 9.34, complete the operation table for the group G in the case when b²=a².

Classify all groups of the given order:143

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