That's a typo for "of" or nothing.

Student or Learner
The phrase "killing a something" appears to be unusual to me. It is normally either "killing something" or "killing a thing" and thus Cambridge Dictionary has made a mistake. I am not absolutely sure.
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decimation
noun [ U ]
UK /ˌdes.ɪˈmeɪ.ʃən/ US /ˌdes.əˈmeɪ.ʃən/
the act of killing a something in large numbers, or reducing something severely:
the virtual decimation of the population through influenza.
Source: https://dictionary.cambridge.org/dic...?q=decimation+
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Oxford Dictionaries defines it very clearly:
decimation
Pronunciation /ˌdesəˈmāSH(ə)n/ /ˌdɛsəˈmeɪʃ(ə)n/
NOUN
1The killing or destruction of a large proportion of a group or species.
‘our growing hunger for fish has resulted in the decimation of fish stocks’
That's a typo for "of" or nothing.
I am not a teacher or a native speaker.
It's definitely not a typo for "of". "The act of killing of something" isn't grammatical. The word simply shouldn't be there at all. It should read "The act of killing something ...".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
It is a typo, and I agree it's more likely to be "the act of killing something", but while "the act of killing of something" sounds awkward as a phrase, "the killing of something" works (grammatically) as a general pattern.
The killing of innocent civilians was deemed a war crime.
I am against the killing of animals and all types of hunting.
The latest eruption of violence was sparked by the killing of a local gang leader.
Absolutely. "the killing of something" is fine. "the act of killing of something" is ungrammatical.
Last edited by emsr2d2; 28-Oct-2020 at 14:44. Reason: Fixed typo. Thanks.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
Again, while I agree that the above phrase sounds odd, I don't think the pattern is ungrammatical. Consider:
The act of killing of one's own mother is called matricide.
Of course, it's much more natural and straightforward to simply say "The act of killing one's own mother is called matricide", but I don't think the above sentence is ungrammatical. It maybe a little awkward, but it's not ungrammatical in my opinion.
Well, you're the English teacher, but I can't see a reason for throwing an extra of into that sentence.
Not a professional teacher
It is a matter of considering the possibilities instead of the typo 'a'. "Of" may not be very natural but is possible.
I am not a teacher or a native speaker.