Az egyesület képzése ingyenes.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since egyesület begins with a vowel sound, the correct form is az egyesület.

So:

• a ház = the house
• az egyesület = the association

Egyesület means association, society, or sometimes club, depending on context.

It usually refers to an organized group, often a non-profit or membership-based organization.

So az egyesület means the association.

This is a standard Hungarian possessive structure.

Hungarian usually does not use a separate word like of or an apostrophe like English ’s. Instead:

• the possessor comes first: az egyesület = the association
• the thing possessed gets a possessive ending: képzése

So:

• az egyesület képzése = the association’s training / the training of the association

This is very common in Hungarian:

• a cég vezetője = the company’s leader
• a fiú könyve = the boy’s book

The ending -e is a third-person singular possessive suffix.

The base word is képzés, which means training, course, or education/training program, depending on context.

When you want to say his/her/its training or the association’s training, Hungarian adds a possessive suffix to the noun:

• képzés = training
• képzése = his/her/its training or the association’s training

Here the possessor is az egyesület, so képzése means its training = the association’s training.

In Hungarian, the verb to be is often omitted in the present tense, third person, when the predicate is a noun or adjective.

So English:

• The association’s training is free

Hungarian:

• Az egyesület képzése ingyenes.

There is no visible word for is, but it is understood.

Compare:

• A ház nagy. = The house is big.
• A könyv érdekes. = The book is interesting.

This omission is completely normal.

Because ingyenes is an adjective, while ingyen is usually an adverb.

• ingyenes = free, free of charge
• ingyen = for free, free of charge in an adverb-like way

In this sentence, ingyenes describes képzése, so an adjective is the natural choice:

• Az egyesület képzése ingyenes. = The association’s training is free.

You may also hear ingyen in other structures, but ingyenes fits this sentence best.

Képzés is a broad word. Depending on context, it can mean:

• training
• course
• program
• instruction
• sometimes education/training process

So Az egyesület képzése ingyenes could be understood as:

• The association’s training is free
• The association’s course is free
• The association’s training program is free

The best English choice depends on the wider context.

It is singular.

• képzése = its training / its course
• képzései = its trainings / its courses

So this sentence refers to one training/course/program in a general or collective sense, not several separate ones.

In Hungarian possessive phrases, the article usually appears before the possessor, not separately before the possessed noun.

So Hungarian says:

• az egyesület képzése

not something like:

• az egyesület a képzése

This whole phrase functions as the association’s training.

That is normal Hungarian structure.

Yes. Hungarian word order is flexible, and changing it often changes the focus or emphasis.

The version you have:

• Az egyesület képzése ingyenes.

is a neutral statement: The association’s training is free.

If you say:

• Ingyenes az egyesület képzése.

the meaning is still basically the same, but now ingyenes is more emphasized, like:

• The association’s training is free.
• It’s free, the association’s training.

So the original order is the most neutral one.

A useful way to mentally break it down is:

• az egyesület = the association
• képzés-e = training-its
• ingyenes = free

So, very literally:

• the association training-its free

Of course, natural English is:

• The association’s training is free.

This kind of literal breakdown can help you get used to Hungarian possessive structure.