Chapter 5.3, Problem 1E

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

1. Claim. If H and K are subgroups of a group G, then H n K is a subgroup of G.
   "Proof." Let $a,b\in H\cap K.$ Then $a,b\in H$ and $a,b\in K.$ Because H and Kare subgroups, $a{b}^{-1}\in H$ and $a{b}^{-1}\in K.$ Therefore, $a{b}^{-1}\in H\cap K.$
2. Claim. If H is a subgroup of a group G and $x\in H,$ then $xH=\left\{xh:h\in H\right\}$ is a subgroup of G.
   "proof." First, the identity $e\in H.$ Thus, $x=xe\in xH.$ Therefore, $xH\ne \varnothing .$ Second, let $a,b\in xH.$ Then $a=xh$ and $b=xk$ for some $h,k\in H.$ Then we have $a{b}^{-1}=\left(xh\right){\left(xk\right)}^{-1}=\left(xh\right)\left(k^{-1}\right)\left(x^{-1}\right)=x\left(h{k}^{-1}{x}^{-1}\right)\in xH.$ Therefore, $xH$ is a subgroup of G.