Let $G$ be a group with order 12. Which of the following claims are false?

Does Lagrange's Theorem imply that there could exist a subgroup with order 6? I'm not sure where to begin. The question was taken from a past test with multiple choice answers, and the question asked which of the following claims were false:

• G must have an element of order 2.
• G must have a subgroup of order 6.
• G must have a subgroup of order 2.
• None of the above, they are all true.

  • 1
    $\begingroup$ How is it a "solution" if it does not justify its claim (when the only problem here that needs solving is why the claim is true). Also no, Lagrange theorem only tells you that $G$ cannot have a subgroup of order $5$, $7, 8, \ldots, 11$. $\endgroup$ – M. Vinay Mar 19 at 2:14
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    $\begingroup$ Several groups of order 12 have subgroups of order 6. Is the problem translated incorrectly? $\endgroup$ – Randall Mar 19 at 2:15
  • $\begingroup$ You must mean that "$G$ may not have a group of order $6$". Here, the "may not" is not imperative — it does not mean "must not". It means, "It is not necessary that $G$ has a subgroup of order $6$". $\endgroup$ – M. Vinay Mar 19 at 2:16
  • $\begingroup$ Sorry for the confusion, I made an edit to the original post. $\endgroup$ – jd94 Mar 19 at 2:17
  • $\begingroup$ The statement that the statement "$X$ must be true" is false does not say that "$X$ must be false". It means "$X$ may (or may not) be true". $\endgroup$ – M. Vinay Mar 19 at 2:18

The first option is true because of Cauchy's theorem. This will also imply the existence of subgroup of order $2$(consider the subgroup generated by element of order $2$). Second option is false because $A_4$ is a group of order $12$ which has no subgroup of order $6$.


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