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I have a problem with using "more than half".
Here are two ways of using "more than half" in two phrases:
More than half the group
More than half of jobs
In my view, we can use "of" when the noun that comes after "more than half" is a plural noun. So we should not use "of" before the singular noun. Am I right?
You are partially correct. In the sentence "more than half the group," the word "of" is implied, so technically the sentence is "more than half of the group." In the second sentence you provided, you could omit "of," but you would have to include the article "the" (so it would become "more than half the jobs"). The omission of "of" is not specific to singular or plural necessarily. "More than half the pizza" and "more than half the pizzas" are both colloquially correct. To reiterate, the word "of" is implied*, even when it is omitted, and this phenomenon is not specific to plural or singular.
*This omission may have come from Latin, in which the genitive case does not have an auxiliary word meaning "of" like English. English based many of its strange grammar rules off of Latin. This MAY BE one of those instances.
Example of Latin genitive: Filius Dei
Example of English genitive: Son of God
