# semi direct of quaternionic group [duplicate]

## marked as duplicate by Dietrich Burde, Najib Idrissi, Namaste group-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Sep 22 '14 at 12:33
The only non trivial subgroups of $\mathbf{Q}$ are $\{1,-1\}$, $\{i\}$, $\{j\}$ and $\{k\}$. Since one needs at least one group of order $2$ in the decomposition $\mathbf{Q}=GH$ if must be $\{1,-1\}$, but this group is the center of $\mathbf{Q}$ so can not be a factor of the decomposition.