Abstract: The integer sequences with first term $1$ comprise a group $\mathcal{G}$ under convolution, namely, the Appell group, and the lower triangular infinite integer matrices with all diagonal entries $1$ comprise a group $\mathbb{G}$ under matrix multiplication. If $A\in \mathcal{G}$ and $M\in \mathbb{G},$ then $MA\in \mathcal{G}.$ The groups $%% \mathcal{G}$ and $\mathbb{G}$ and various subgroups are discussed. These include the group $\mathbb{G}^{(1)}$ of matrices whose columns are identical except for initial zeros, and also the group $\mathbb{G}^{(2)}$ of matrices in which the odd-numbered columns are identical except for initial zeros and the same is true for even-numbered columns. Conditions are determined for the product of two matrices in $\mathbb{G}^{(m)}$ to be in $\mathbb{G}%% ^{(1)}.$ Conditions are also determined for two matrices in $\mathbb{G}%% ^{(2)}$ to commute.