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