Κάθε φοιτητής γράφει το όνομά του στο τετράδιό του.

In Greek, κάθε means each / every and is normally followed by a singular noun.

So:

• κάθε φοιτητής = each student / every student
• not κάθε φοιτητές

This is similar to English every student, which also uses a singular noun.

φοιτητής is in the nominative singular because it is the subject of the sentence: the one doing the writing.

The basic structure is:

• Κάθε φοιτητής = subject
• γράφει = verb
• το όνομά του = direct object
• στο τετράδιό του = prepositional phrase

Because the subject is grammatically singular: κάθε φοιτητής.

Even though the idea refers to many students in general, Greek treats κάθε φοιτητής as one student at a time, so the verb is singular:

• Κάθε φοιτητής γράφει...
• not Κάθε φοιτητής γράφουν...

γράφει is the present tense.

Depending on context, the Greek present can mean:

• writes
• is writing

In this sentence, because of κάθε, it is most naturally understood as a general / habitual action:

• Every student writes his/her name in his/her notebook.

So here writes is the best way to think about it.

Greek usually uses the definite article with nouns much more often than English does.

So Greek naturally says:

• το όνομά του = his name
• literally, the name of him

This is the normal Greek pattern:

• το βιβλίο μου = my book
• η μητέρα της = her mother
• το σπίτι τους = their house

Here του is a weak genitive pronoun meaning his, and in many contexts it can also function like his own.

So:

• το όνομά του = his name
• literally, the name of him

In a sentence with κάθε φοιτητής, this is understood distributively:

• each student writes his own name

Greek often uses this simple possessive form where English might emphasize own.

Because there are two different possessed nouns:

• το όνομά του = his name
• στο τετράδιό του = in his notebook

Greek repeats the possessive pronoun when both nouns are possessed by the same person. English can sometimes do the same, and Greek normally does not omit it here.

So the sentence is literally structured like:

• Every student writes his name in his notebook.

That is the normal Modern Greek pattern with these short possessive forms:

• το όνομά μου = my name
• το όνομά σου = your name
• το όνομά του = his name

So Greek usually says:

• article + noun + weak genitive pronoun

This is different from English, where the possessive comes before the noun:

• my name
• his notebook

στο is a contraction of:

• σε + το = στο

Here:

• σε = in / to / at depending on context
• το = the

So:

• στο τετράδιό του = in his notebook

This contraction is extremely common in Modern Greek:

• στο = σε το
• στη = σε τη(ν)
• στον = σε τον

In Modern Greek, the preposition σε normally takes the accusative.

So after σε / στο, you use the accusative form:

• στο τετράδιο
• στο σπίτι
• στο βιβλίο

That is why you get στο τετράδιό του.

This happens because του is an enclitic: a weak word that leans on the word before it.

When a Greek word is stressed on the antepenultimate syllable and is followed by an enclitic such as μου, σου, του, της, μας, σας, τους, Greek often adds a second written accent to show the stress pattern more clearly.

So:

• όνομα → το όνομά του
• τετράδιο → το τετράδιό του

This is a standard spelling rule in Modern Greek.

Grammatically, φοιτητής is a masculine singular noun and most literally means male university student.

However, depending on context, Greek has often used the masculine form in a generic way, especially in textbooks or general statements. So the sentence may mean:

• a male student specifically, or
• every student in a general sense

If someone wanted to be fully explicit about both genders, they might say something like:

• κάθε φοιτητής και φοιτήτρια

No, Greek word order is fairly flexible, though this sentence is a very natural neutral order.

The given order:

• Κάθε φοιτητής γράφει το όνομά του στο τετράδιό του.

is straightforward and standard.

Other word orders are possible for emphasis, but they may sound more marked. For a learner, the given version is the best basic model:

• subject + verb + object + prepositional phrase

Because Greek normally uses the ordinary possessive clitic του when the meaning is clear from context.

So:

• το όνομά του naturally means his name / his own name
• το τετράδιό του naturally means his notebook / his own notebook

A stronger reflexive expression exists, such as τον εαυτό του, but that is used in different structures, usually for himself as an object, not for ordinary possession.

So in this sentence, simple του is exactly what Greek normally uses.