- #1

mathusers

- 47

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any suggestions on how to go about doing these?

**INFORMATION**

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if K,Q are groups [itex] \varphi : Q \rightarrow Aut(K) [/itex] is a homomorphism the semi direct product [itex] K \rtimes_{\varphi} Q[/itex] is defined as follows.

(i) as a set [itex] K \rtimes_{\varphi} Q = K \times Q[/itex]

(ii) the group operation * is [itex](k_1,q_1)*(k_2,q_2) = (k_1 \varphi(q_1)(k_2),q1q2)[/itex]

**THE QUESTION**

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Verify formally that [itex] K \rtimes_{\varphi} Q = (K \times Q, *, (1,1)[/itex] is a group and find a formula for [itex](k,q)^{-1}[/itex] in terms of [itex]k^{-1},q^{-1}[/itex] and [itex]\varphi[/itex]

-----> to show that it is a group, i know i have to show that the 4 conditions for being a group (e.g. associativity, closure, existence of identity element, existence of inverse) have to be satisfied. but not really too sure how to show it.. and I am completely baffled for the 2nd part of the question.

please help out :) thnx