Which of the following statements regarding `1,3`- dimethl cyclobutane is/are correct?

Options

A. The cis has plane of symmetry, while the trans form has both plane and centre of symmetry.
B. The cis form has both plane and centre of symmetry, while the trans form has only plane of symmetry.
C. Both have only plane of symmetry.
D. Neither of them has any element of symmetry.

Answer

To determine which statements regarding 1,3-dimethylcyclobutane are correct, we need to analyze the structure of the compound and the definitions of symmetry elements.

### Step 1: Understand the Structure of 1,3-Dimethylcyclobutane
1,3-Dimethylcyclobutane is a cyclobutane ring with two methyl groups attached at the 1 and 3 positions. The structure can exist in two geometric isomers: cis and trans.

### Step 2: Identify the Geometric Isomers
- **Cis-1,3-Dimethylcyclobutane**: Both methyl groups are on the same side of the cyclobutane ring.
- **Trans-1,3-Dimethylcyclobutane**: The methyl groups are on opposite sides of the cyclobutane ring.

### Step 3: Analyze the Symmetry Elements
1. **Cis Isomer**:
- **Plane of Symmetry**: The cis isomer has a plane of symmetry because if you were to draw an imaginary line through the middle of the molecule, the two halves would be mirror images of each other.
- **Center of Symmetry**: The cis isomer does not have a center of symmetry because there is no point in the molecule from which equal groups are present at equal distances in opposite directions.

2. **Trans Isomer**:
- **Plane of Symmetry**: The trans isomer has a plane of symmetry as well.
- **Center of Symmetry**: The trans isomer also has a center of symmetry because there exists a point in the molecule from which equal groups are present at equal distances in opposite directions.

### Step 4: Evaluate the Statements
1. **Statement 1**: "Cis has a plane of symmetry while trans has both plane and center of symmetry."
- **Correct**: This statement is true as explained above.

2. **Statement 2**: "Cis form has both plane and center of symmetry while trans form has only plane of symmetry."
- **Incorrect**: The cis form only has a plane of symmetry, not a center of symmetry.

3. **Statement 3**: "Both have only plane of symmetry."
- **Incorrect**: The trans form has both plane and center of symmetry.

4. **Statement 4**: "Neither of them has any element of symmetry."
- **Incorrect**: Both isomers have symmetry elements.

### Conclusion
The only correct statement regarding 1,3-dimethylcyclobutane is the first one. Therefore, the answer is option number 1.