[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the orthogonal easy groups are

[[math]] \xymatrix@R=30pt@C=80pt{ H_N\ar[r]&O_N\\ S_N'\ar[u]&B_N'\ar[u]\\ S_N\ar[r]\ar[u]&B_N\ar[u]} [[/math]]

where [math]S_N'=S_N\times\mathbb Z_2[/math] and [math]B_N'=B_N\times\mathbb Z_2[/math].