Bettas have a reputation for aggression. And the reputation is partially deserved. Males bettas are extremely aggressive towards other male bettas. However, they do get along with other fish well, so you can keep them in a 10-gallon aquarium with other species that will not harm the betta.
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Tetras work well with bettas and work well in small tanks. Most tetras have a natural tendency to school and should be kept in groups of five to six of their own species. In smaller groups, tetras tend to feel stress and pine away. Several small, colorful tetras that are well-suited to the betta tank are the neon tetra, the black phantom tetra and the lemon tetra. These tetras are small and peaceful, and they would make great tank mates for a betta in a 10-gallon tank.
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Corydoras catfish, or cory cats for short, act more like tetras than most catfish. While they do hang out in the bottom of the tank and scavenge uneaten food, they school like tetras. Dozens of species are available, and have a variety of colors and patterns. In many ways, they complement bettas. They tend to have drabber colors (though striking patterns), occupy the bottom of the tank and hang out in groups. This contrast complements bettas, and their peaceful demeanor and small size allows them to share an aquarium easily.
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The cichlids family includes many varied fish. While some cichlids are known for aggression, some smaller members of the family get along well with other fish. The ram and the krib both stay small and rarely bother other fish. Both of these cichlids come in several colors, hang out near the bottom of the tank, and may even spawn if you have a pair. Both the ram and the krib can work well with a betta in a 10-gallon tank.
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Dwarf gouramis come from the same family of fish as the betta. While gouramis in general also have a reputation for aggression, dwarf gouramis tend to be much more docile. These fish stay small, possess bright colors and get along well with other fish. The have much in common with bettas, requiring peaceful tank mates and tolerating beginners' mistakes well. There are several species, and several color varieties of each. One or two will easily share a tank with a betta.