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We are familiar with properties and structure of topological spaces. One ofthe powerful tools, which help us to figure out the structure of topologicalspaces is (Leray- Serre) spectral sequence. Although Eilenberg-Maclane spaceplays important roles in topology, and respectively geometry. Actually findingcohomology groups of this space can be useful for classifying spaces, and alsohomotopy groups structure of these groups. This paper discusses how to computecohomology groups of Eilenberg-Maclane spaces $K(\mathbb{Z}, 4)$ and$K(\mathbb{Z}, 5)$ (cohomology degree less than $11$). Furthermore we give themethod to find cohomology groups of $K(\mathbb{Z}, n)$. Some proofs are givento the basic facts about cohomology group of $K(\mathbb{Z}, 5)$ and$K(\mathbb{Z},4).$

Cohomology Group of $K(\mathbb{Z},4)$ and $K(\mathbb{Z},5)$