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In a primary (1°) halogenoalkane, the carbon which carries the halogen atom is only attached to one other alkyl group.
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For $\ce{CH3Cl}$, the carbon which carries the halogen atom has no other alkyl groups attached, yet my class notes say "the alkyl halide has a primary centre". So is a primary alkyl group one with is attached to 0 or 1 other alkyl group and not just 1?

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Yes, you are correct.

The IUPAC defines alkyl groups in the following manner:

Univalent groups derived from alkanes by removal of a hydrogen atom from any carbon atom $\ce{–C_{$n$}H_{$2n+1$}}$. The groups derived by removal of a hydrogen atom from a terminal carbon atom of unbranched alkanes form a subclass of normal alkyl (n-alkyl) groups $\ce{H(CH2)_{$n$}}$. The groups $\ce{RCH2}$, $\ce{R2CH~(R \neq H)}$, and $\ce{R3C~(R \neq H)}$ are primary, secondary and tertiary alkyl groups, respectively.

Note that for primary alkyl groups hydrogen may be a remainder, i.e. $\ce{R $=$ H, Alkyl, Aryl}$, therefore any halogenoalkane with the structure $\ce{RH2C-X}$ is primary.

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