Correct spelling, explanation: management is the correct spelling because the core word here is manage to which we added the noun suffix -ment. The is no need to skip e here as the suffix starts with a consonant (m), not a vowel. Therefore managment is incorrect and management is the correct form.

Definition of management:
noun, an action of managing, organising something or someone; person or group of people who manage something or someone
They had terrible management in their company and soon they went bankrupt.
Management of our department decided to start the project next week.

Incorrect spelling, explanation: Despite the fact, that pronunciation points to no e, management is spelled with this vowel – the e is silent. This is due to the presence of this vowel in the core word – verb manage. We don’t drop the e in management because the suffix -ment starts with a consonant. This is why managment is an incorrect spelling.

Incorrect spelling

Incorrect spelling

Incorrect spelling

Incorrect spelling

Incorrect spelling