<p>"That Stalin actually murdered Kirov, his popular second in command, remains a mystery, but Stalin's Great Purges prove their ruler had the power to do so."</p>
<p>"That Stalin actually murdered Kirov, his popular second in command, remains a mystery" doesn't sound right to me; it doesn't seem to be able to stand alone. Still, its the correct answer from Kaplan.</p>
<p>“That Stalin actually murdered Kirov, his popular second in command, remains a mystery, but Stalin’s Great Purges prove their ruler had the power to do so.”</p>
<p>You can ignore “his popular second in command” because it is a prepositional phrase. Then it reads, “That Stalin actually murdered Kirov remains a mystery” seems to be a fragment to me. </p>
<p>I think the best revision would be to replace “That” with “Whether”. </p>
<p>The face that “their” not referring to anyone is not so bad. “Their ruler” was probably defined in the previous sentence, but you don’t know. This is the same reason that you don’t refer to John in every sentence. You don’t want to sound redundant. Therefore, we occasionally use he, him, my friend, etc.</p>
<p>You’ve got to watch out for Kaplan material as it is riddled with mistakes.</p>