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Every Group of Order 4 is Cyclic or of Klein’s Type: An Insight into Group Theory

January 07, 2025Science2321
How to Prove That Every Group of Order 4 is Either Cyclic or of Kleins

How to Prove That Every Group of Order 4 is Either Cyclic or of Klein's Type

To show that every group of order 4 is either cyclic or of Klein’s type, we can utilize group theory concepts. Specifically, we will follow a structured approach to understand and prove this statement.

Step 1: Using Lagranges Theorem

According to Lagrange’s Theorem, in a finite group (G), the order of every subgroup must divide the order of the group. For a group (G) of order 4, the possible orders of its subgroups are 1, 2, or 4.

Step 2: Identifying Possible Subgroup Structures

1. Trivial Subgroup: Every group has a trivial subgroup, denoted by ({e}), with order 1.

2. Subgroups of Order 2: By Lagrange’s theorem, if (G) has a subgroup of order 2, it must have at least one such subgroup as 2 divides 4. Let this subgroup be denoted by (H). The element (g otin H) must then also exist and it must generate the group along with (H).

Step 3: Analyzing the Group Structure

Cyclic Group (C_4): If (G) is cyclic, it can be generated by a single element (g) such that (G {e, g, g^2, g^3}). This group is isomorphic to (mathbb{Z}/4mathbb{Z}).

Kleins Four-Group (V_4): If (G) is not cyclic, it must have multiple elements of order 2. The only non-cyclic group of order 4 is the Klein four-group, which can be represented as (V_4 {e, a, b, ab}) where (a^2 e), (b^2 e), and (ab^2 e). This group has the property that every non-identity element has order 2.

Step 4: Conclusion

Therefore, any group (G) of order 4 must either be cyclic (isomorphic to (mathbb{Z}/4mathbb{Z})) or isomorphic to the Klein four-group (V_4).

Summary

- If (G) is cyclic, it is isomorphic to (mathbb{Z}/4mathbb{Z}).

- If (G) is not cyclic, it has three elements of order 2 and is therefore isomorphic to the Klein four-group (V_4).

This concludes our demonstration that every group of order 4 is either cyclic or of Klein’s type.

Further, if (e) denotes the identity element of (G), there are two possibilities for the order of the other elements of (G):
- (o(g) 2) for each (g in G) and (g e e)
- (o(g) 4) for some (g in G)

In case (i), if (abc in G setminus {e}), then (a^2 b^2 c^2) and (ab c), because (ab e ab) and (ab e) implies (a ae aab a^2b b). Thus, we may identify (G) with (mathbb{Z}_2 oplus mathbb{Z}_2) with (e leftrightarrow 00), (a leftrightarrow 10), (b leftrightarrow 01), and (c leftrightarrow 11). This identification is an isomorphism between the group (G) and the Klein 4-group (mathbb{Z}_2 oplus mathbb{Z}_2).

In case (ii), (G) is cyclic and thus (G cong mathbb{Z}_4).

Conclusion: Thus, we have shown that any group of order 4 must be either cyclic or of Klein's type, thereby solidifying the understanding of group structures of this order.