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Question 1 Here's a very simple example. Let $Q= < i,j,k > $ be the quaternion group. $-1$ is the unique involution, so is in its own conjugacy class. $(Q,-1)$ is nonsplitting, and $Q$ is its own Sylow 2-subgroup. In general, take any finite group with a unique involution. (These turn out to be cyclic, quaternion and 2 other kinds.) I don't know what happens when you take a group with more than one involution. |
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Here's a very simple example. Let $Q= < i,j,k > $ be the quaternion group. $-1$ is the unique involution, so is in its own conjugacy class. $(Q,-1)$ is nonsplitting, and $Q$ is its own Sylow 2-subgroup. In general, take any finite group with a unique involution. (These turn out to be cyclic, quaternion and 2 other kinds.) I don't know what happens when you take a group with more than one involution. |
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