I read that the black hole at the center of the galaxy has much less mass than the galaxy itself and that it is somehow held together by dark matter.
So now I am wondering if it is possible that there are galaxies with a star at the center?
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Galaxy nuclei are full of stars. Any star passing by will exchange momentum with the central star and will perturb its position. Stars of similar mass will be able to completely eject the central star from its privileged position.
Massive stars (and it has to be massive, see p. 1) tend to live few million years and die, forming (in the general case) a stellar-mass black hole. On the other hand, galaxies live for bilions of years. So we end up with a very young galaxy with a small black hole in the center.
Massive stars, as well as their remnant black holes, tend to sink towards the center of the clusters of stars (and the galaxy nucleus has any reason to possess the same dynamics). The object being a massive star is not of great importance, because they quickly (on the timescale of the galaxy formation) convert to black holes.
In short, the modern consensus is that galaxies do have a black hole in the center. But even if we start with a galaxy that does not have one (we know that galaxy mergers sometimes eject the central black holes), it will grow its brand new central black hole soon.
The short answer is, not really. Galaxies are unlikely to have a star at the "center".
When you say a star is at the center, what you're really trying to say is that that star is the gravitationally dominant object at the center of the galaxy. However, almost all galaxies have a supermassive black hole (SMBH) at the center. Why this is the case is still not 100% worked out, but the general consensus is that SMBH are crucial for the formation of galaxies, and may in fact form alongside the rest of the galaxy. What that means is that you're very unlikely to see a galaxy that doesn't have a SMBH in the center because the SMBH is so crucial to the very existence of that galaxy.
Trying to replace a SMBH with just a star (even a massive one) won't cut it because SMBH's usually have a mass equal to millions of solar masses. Even a 100 solar mass star (which, by the way, wouldn't live very long) just isn't enough to support the same dynamics that a SMBH can when it comes to galaxy formation.
it is somehow held together by "Dark matter"
Just to address this tangential point, black holes are not held together by Dark Matter. You possibly may have heard that galaxies themselves are held together by Dark Matter and are just misremembering. Black holes are held together by their own gravitational force of all the mass that they have "eaten" over their lifetime.
Galaxies are gravitationally bound by the mass of all the stars, gas, dust, planets, black holes and "dark matter" that they are made of. In fact the majority of the mass comes from the "dark matter" (which is a little problematic, as we don't know what it actually is)
If there is a large black hole in a galaxy, as it passes by the stars in the galaxy it will tend to transfer kinetic energy to the stars, and so will lose energy itself, and move towards the centre of the galaxy. Over time, all the large black holes will gather and merge at the centre of the galaxy, forming a "supermassive black hole". This is why most large galaxies seem to have a large black hole at their centre.
Galaxies have black holes at their centre, because large black holes will tend to move towards the centre of a galaxy, not because the galaxy needs to have anything in the centre to keep it spinning.
There are some, like the triangulum galaxy that don't seem to have a large black hole at their centre. This probably means that there is actually an intermediate-sized black hole there. Even if not, that would probably mean that there is nothing that is at rest at the centre of the galaxy. There is no need for there to be anything at the centre. The mass of the stars, gas, and dark matter is enough to bind the galaxy together and keep it spinning without anything at the centre.
We have discovered that most galaxies have a supermassive black hole at their centre. So we suspect that galaxy formation itself usually involves a central black hole. Or one forms and gradually ends up migrating towards the centre if it didn't start there. Something, somehow, anyway.
But the black hole itself, huge though it may be, is not capable of single handedly creating a galaxy. It didn't start the galaxy off. It might be millions or even a few billions of times the size of our sun, but for example, our Milky Way galaxy has about 100-400 billion stars in it, spread far enough that the central black hole just isn't a big deal to most of it. Our entire central black hole could vanish with almost no change to the galaxy. You wouldn't even notice if you didn't go looking hard in that specific area.
So if you want to find what * really * started a galaxy growing, and keeps it together, its not black holes, or even a supermassive black hole, or anything at the center of the galaxy. You have to look elsewhere, far spread out in space, for the answer.......
We can "weigh" our galaxy and others, by measuring how they turn (spin, rotate) in space. If you imagine swinging something on the end of a string, the faster you spin it the more force you need to use. In the same way, we can measure how strongly stars are pulling towards each other, by how fast they move and the galaxy "spins".
Crucially, our galaxy is not at all like our solar system, where one huge central object (the Sun) mostly holds everything else in place.
The pulling is the galaxy as a whole, it doesn't mean its all some central object doing it. Everything in the galaxy pulls towards everything else, whether it's stars, gas clouds, black holes, planets, or anything else. Its that combined pull that holds it together and lets it stay together while it turns.
The problem is, when you do that calculation, you find that there just isnt enough stuff to hold the galaxy together and make it turn as we see it turning. In round numbers, the total mass of all the stars, gas clouds, planets and visible matter in our galaxy is about 50,000,000,000 times the mass of our sun.
But to rotate as it does, the total mass would actually have to be about twenty times as much as that.
Something's wrong. There's actually about 1,000,000,000,000 times the mass of the sun hiding in and around our galaxy that we can't even see. We can be pretty sure from theoretical workings, that its not huge numbers of tiny clouds, invisible stars, or made up of black holes. Its some new kind of matter that is "dark" (not able to be seen) to our usual instruments. Until we know what this new stuff is,we will call it "Dark Matter".
We can work out quite a lot about dark matter, not just how much of it there is, but what kinds of things it could be. The most likely candidate is a new type of subatomic particle that only weakly responds to gravity, and not at all to most other forces. Because it doesn't interact with other forces, we don't see it directly in our instruments. Because it interacts with gravity, it can shape our galaxy and lead to galaxy formation.
We now believe that this new "dark matter" actually led to galaxy formation in the first place, in the early universe.
When matter pulls in on itself like in a star, it gets very hot, and the heat stops it collapsing easily. Like how our sun doesn't collapse. The catch is, to collapse a bit (or at all), it needs to react to gravity. To collapse a lot, it has to be able to get rid of all the heat (or other forms of energy) which that produces. Dark matter can't do that though, because it just doesn't interact in any ways that let it ditch much heat. So it kind of collapsed to vast thin clouds, and couldn't go further. But once those vast thin clouds formed, they attracted "ordinary" matter which could collapse further by losing heat (or other forms of energy), and that's how galaxies began.
A central black hole often seems to also form, but fundamentally its the dark matter that started it all, long, long, before.
Since I read that the black hole in the center of the galaxy has much less mass than the galaxy itself and that it is somehow held together by "Dark matter".
These are both correct. See above for details. Ask if any things unclear.
so now I am wondering if it is possible that there exist galaxies where a star is at the center of it.
It wouldn't matter. A galaxy doesn't "need" to have a single object at or near its center. Most do, and its a black hole, but there doesn't have to be anything there as far as we know, and some don't. Even the biggest stars are tiny and microscopic compared to a galaxy.
So it would be completely unimportant if a star was at the center, for some galaxy or another, it wouldn't make any difference. A central object, however big, does not holds a galaxy together or makes it turn.
A second problem is, if you want "a star" at the center, then presumably you want a biiiiiig star. Big stars die really fast, so it wouldn't have survived millions, let alone billions of years, that the galaxy has existed. It would be gone long long before the galaxy even formed. And even the very biggest stars are tiny compared to the supermassive black holes we know of.
In is not possible for a star to be (maintain a stable position) in the center of a galaxy.
For any one object to be (in) the center of a dynamic collection of anything, it has to somehow 'belong' there, like the bubbles in your coffee when you stir it. They didn't just get there by chance.
For something to belong in the center of a dynamic collection, there can be three reasons.
It can be there by chance, just passing through so to speak
It can be carried and held there by the dynamics of the collection, like the bubbles in the coffee.
It can be what is holding the collection together, by being the biggest thing around, like the sun in our solar system.
For a star to travel through the center of a galaxy by chance might be a possibility, but it wouldn't be there by either one of the other reasons. It is not unique enough to be the only thing there and it's not big enough to be what holds the galaxy together.