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This is the mass spectrum for 4-methylpentan-1-ol:

mass spectrum for 4-methylpentan-1-ol

From its structural formula, what could be the fragmentation which results in a peak at $m/z = 56?$

I was considering a few possibilities; could it be $\ce{C3H4O+}?$

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Unfortunately I did not find a high resolution spectrum of this compound, which would have quickly answered your question.

Nevertheless, ionized alcohols and even more primary alcohols have a main fragmentation pathway which is the loss of water to yield an ionized alkene. In this case, the corresponding alkene would be 4-methyl-1-pentene, at m/z 84. The peak is present although rather small in the EI spectrum. From there, it is quite obvious, when one looks at the fragmentation spectrum of 4-methyl-1-pentene (NIST Webbook spectrum), that it shares a lot of similarities with the 4-methylpentan-1-ol EI spectrum. (See for instance peaks at m/z 69, 56, 43, 42, 41.) From there, it is quite likely that the m/z 56 peak is ionized isobutene ($\ce{C_4H_8}^{\bullet +}$) arising from a retro-ene type rearrangement.

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