Definition 1: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.

Definition 2: For two matrices $A$ and $B$, the Hadamard product is defined as $(A \circ B)_{i,j} = A_{i,j}B_{i,j}$.

Definition 3: An Hadamard group (HG for short) $G=\{J,H_1,H_2,...,H_m\}$ is a matrix group under Hadamard product $\circ$, where

$J$ is the all-ones matrix,

$H_1$,$H_2$,...,$H_m$ are $m$ $n$-by-$n$ Hadamard matrices.

Questions:

(1) Is there any result or conclusion about HG?

(2) Given $n$, what is the maximum value of $m$?

(3) Given $n$, how to construct a non-trivial example of HG?