It is important to me for her to study English.
Does this sentence make sense?
I understand what it means but it's not the most natural way to say it and I think some people might be confused by it. I would try "It is important to me that she study English" (note you will also hear "that she studies English" here).
Will it be grammatical if we say '...that she studies English'?
We don’t use the subjunctive very much in contemporary English unless we wish to sound very formal. With verbs like suggest, recommend, insist and adjectives like important, essential, imperative, crucial, vital, we often use should + infinitive instead of the subjunctive or we can use the normal tense form. The reporting verbs and adjectives above are normally followed by a that-clause in which that itself is often omitted.
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I know 'should' is omitted between she and study.
Will it be grammatical if we say '...that she studies English'?
What should I answer my students if they claim
they have heard 'It is important to me that she studies'?
I've taught that we should say 'It is important to me that she should study' or
'It is important to me that she study'.
It's not the bare infinitive in 'It is important that she study', though it has the same form; it's the present subjunctive. It is the bare infinitive in 'It is important that she should study'.I always tell my students that if they can remember to use the bare infinitive then that's great and I always remind them of it but I would not tell them "that she studies" is incorrect. Having said that, it's impossible to tell what the people marking a test or an exam think so in a formal exam situation I advise them to use the bare infinitive. I don't advise them to use "that she should study".