Originally Posted by

**joanne_q** hi a little help would be kindly appreciated here guys.

any suggestions on how to go about doing these?

**INFORMATION**

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

(i) as a set $\displaystyle K \rtimes_{\varphi} Q = K \times Q$

(ii) the group operation * is $\displaystyle (k_1,q_1)*(k_2,q_2) = (k_1 \varphi(q_1)(k_2),q1q2)$

**THE QUESTION**

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