# Tour:Factsheet four (beginners)

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To summarize some of the things we have seen so far in part four:

• A finite group can be completely described by its multiplication table: a table where the rows and columns are indexed by the entries of the group, and an entry in a cell is a product of its row and column. Nostalgia moment: We saw in parts one and two of the tour that the binary operation on a group completely determines the identity element and inverse map. This can be seen from the fact that we can look at the multiplication table and visually determine the identity element and inverses.
• An isomorphism between groups is a bijective map that preserves all the group operations. Nostalgia moment: We saw in parts one and two that the binary operation on a group completely determines the identity element and inverse element. Thus, a bijective map that preserves the binary operation automatically preserves the identity element and inverse map.
• We can define a group of integers modulo $n$. A crude way of defining this group is as the set of numbers $\{ 0,1,2,\dots,n-1\}$ where we add two numbers, and if the sum is greater than $n$, subtract $n$ from the sum. A more sophisticated and useful view is to think of an equivalence relation on the group of integers defined by having the same remainder modulo $n$ and consider the group of congruence classes modulo $n$.
• A cyclic group (i.e., a group generated by one element) is isomorphic either to the group of integers, or to the group of integers modulo $n$, where $n$ is the order of that element.
• Any nontrivial subgroup of the group of integers is cyclic on its smallest element. Using this, we can show that any subgroup of a cyclic group is cyclic. In fact, a cyclic group of order $n$ has exactly one subgroup of order $d$ for every $d$ dividing $n$: the subgroup of multiples of $n/d$.
• The subgroups of the group of integers are ordered by inclusion in the same way as their generators are ordered by divisibility. Thus, notions like greatest common divisor and least common multiple for integers correspond to notions of join and intersection of subgroups.
• A group that has no proper, nontrivial subgroup must be cyclic of prime order; conversely any group of prime order must have no proper, nontrivial subgroup. Thus, any group of prime order is cyclic.
• We can consider the congruence classes modulo $n$ under multiplication; these form a monoid, not a group. The invertible elements in this monoid form a group. This group is also the same as the set of elements that additively generate the group of integers mod $n$, and is also the same as the congruence classes mod $n$ that are relatively prime to $n$. The number of such elements is denoted $\varphi(n)$, and is termed the Euler-phi function of $n$.
• The multiplicative group modulo a prime is a cyclic group of order $p-1$. However, there is no formula or easy procedure for finding a generator of this multiplicative group.