any positive integer $m$. There are two non-isomorphic orthogonal groups of a $2m$-dimensional vector space $V$ over the field of $p$ elements with respect to a non-singular symmetric bilinear form $b$ denoted by $GO^+_{2m}(p)$ (if $V$ has a totally isotropic subspace of dimension $m$) and $GO^-_{2m}(p)$ (if $V$ has no totally isotropic subspace of dimension $m$). The orders of these groups are the following: $$|GO_{2m+1}(p)|=2p^{m^2}(p^2-1)(p^4-1)\cdots (p^{2m}-1),$$ $$|GO^+_{2m}(p)|=2p^{m(m-1)}(p^2-1)(p^4-1)\cdots (p^{2m-2}-1)(p^m-1)$$ and $$|GO^-_{2m}(p)|=2p^{m(m-1)}(p^2-1)(p^4-1)\cdots
(p^{2m-2}-1)(p^m+1).$$

For the prime $2$ there are two classes of non-singular symmetric bilinear forms: alternating and non-alternating. Alternating non-singular symmetric bilinear forms are over $2m$-dimensional ${\mathbb{Z}}/(2)$-vector spaces and their orthogonal groups coincide with the symplectic group $\mathrm{Sp}_{2m}(2)$ that has order $$|\mathrm{Sp}_{2m}(2)|=2^{m^2}(2^2-1)(2^4-1)\cdots (2^{2m}-1),$$ (see [@Wi]). The orthogonal group $\mathrm{O}(m,2)$ of a non-alternating non-singular symmetric bilinear form $b$ over an $m$-dimensional ${\mathbb{Z}}/(2)$-vector space $V$ has order $$|\mathrm{O}(2t+1,2)|=2^{t^2}(2^2-1)(2^4-1)\cdots (2^{2t}-1),$$ if $m=2t+1$, and $$|\mathrm{O}(2t,2)|=2^{t^2}(2^2-1)(2^4-1)\cdots (2^{2(t-1)}-1),$$ if $m=2t$ (see [@MW]).

\[ortho\]

Let $p_1$ be an odd prime number. Let $p\neq p_1$ be a prime number. Let $k$ be the order of $p$ in $({\mathbb{Z}}/(p_1))^*$. If $p$ and $k$ are odd, then

-   $p_1$ is a divisor of $|GO_{2m+1}(p)|$ if and only if $k\leq m$,

-   $p_1$ is a divisor of $|GO^+_{2m}(p)|$ if and only if 