How is it possible or legal to issue a warrant for someone just because someone said that you hit them or pushed them?

Courts have held that a single eye-witness to an occurrence is enough to find a person guilty beyond a reasonable doubt of committing a criminal act.  The police need not interview the suspect or advise the suspect that they will issue a warrant.  The standard for the issuance of a warrant is "probable cause", which is one of the lowest burdens of proof in Illinois. If the testimony of one eye-witness is good enough for guilty beyond a reasonable doubt, it can be good enough for probable cause. The fact that the police failed to interview the suspect can be used as a potential cross-examination subject at trial though.  "Officer Humdum, isn't it true you didn't even bother to contact my client to hear his side of the story?  So you came and arrested my client when you only heard one side of the story?" etc.  A hitting or pushing is called a "Battery" in Illinois, which requires unauthorized physical contact of an insulting or provoking nature, or the causing of bodily harm through physical contact.