  We define a new variety of loops we call $\Gamma$-loops. After showing
$\Gamma$-loops are power associative, our main goal will be showing a
categorical isomorphism between Bruck loops of odd order and $\Gamma$-loops of
odd order. Once this has been established, we can use the well known structure
of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy
Theorems for $\Gamma$-loops of odd order, as well as the nontriviality of the
center of finite $\Gamma$-$p$-loops ($p$ odd). Finally, we answer a question
posed by Jedli\v{c}ka, Kinyon and Vojt\v{e}chovsk\'{y} about the existence of
Hall $\pi$-subloops and Sylow $p$-subloops in commutative automorphic loops. By
showing commutative automorphic loops are $\Gamma$-loops and using the
categorical isomorphism, we answer in the affirmative.
