We extend two standard theorems on groups to gyrogroups: the direct product theorem and the cancellation theorem for direct products. Firstly, we prove that under a certain condition a gyrogroup $ G $ can be decomposed as the direct product of two subgyrogroups. Secondly, we prove that finite gyrogroups can be cancelled in direct products: if $ A\cong B $, then $ A\times H\cong B\times K $ or $ H\times A\cong K\times B $ implies $ H\cong K $, where $ A, B, H, $ and $ K $ are finite gyrogroups.
Citation: Teerapong Suksumran. Two theorems on direct products of gyrogroups[J]. AIMS Mathematics, 2023, 8(3): 6278-6287. doi: 10.3934/math.2023317
We extend two standard theorems on groups to gyrogroups: the direct product theorem and the cancellation theorem for direct products. Firstly, we prove that under a certain condition a gyrogroup $ G $ can be decomposed as the direct product of two subgyrogroups. Secondly, we prove that finite gyrogroups can be cancelled in direct products: if $ A\cong B $, then $ A\times H\cong B\times K $ or $ H\times A\cong K\times B $ implies $ H\cong K $, where $ A, B, H, $ and $ K $ are finite gyrogroups.
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Teerapong Suksumran. Two theorems on direct products of gyrogroups[J]. AIMS Mathematics, 2023, 8(3): 6278-6287. doi: 10.3934/math.2023317
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