To evaluate the two statements regarding the order of elements in a group, we need to understand some fundamental concepts from group theory. Statement 1: **If a group has an element of order 15, it must have at least 8 elements of order 15.** **Analysis:** 1. **Order of an element:** The order of an element in a group is the smallest positive integer \( n \) such that \( g^n = e \), where \( g \) is the element and \( e \) is the identity element. 2. **Cyclic Subgroup:** An element of order 15 generates a cyclic subgroup of order 15. By Lagrange's theorem, this subgroup has exactly 15 elements (including the identity). 3. **Counting Elements:** If a group has an element of order 15, it does not necessarily imply that there are at least 8 distinct elements of order 15. The cyclic group of order 15 has a total of 15 elements, but not all of them need to be of order 15. The possible orders of elements in this group can be 1, 3, 5, or 15, depending on their specific properties. Thus, it is possible to have fewer than 8 elements of order 15. For example, in a group of order 15, the only element of order 15 is the generator itself, leading to the conclusion that Statement 1 is **False**. Statement 2: **If a group has more than 8 elements of order 15, it must have at least 16 elements of order 15.** **Analysis:** 1. **Counting Elements Again:** If a group has more than 8 elements of order 15, we need to consider how elements of a given order can behave. 2. **Cyclic Subgroups:** Any element of order 15 generates a cyclic subgroup of order 15, which contains exactly 15 elements. The elements of order 15 in a group will come from distinct cyclic subgroups. 3. **Cosets and Group Structure:** If there are more than 8 elements of order 15, they can come from at least one or more cyclic subgroups of order 15. However, we cannot conclude that for every element of order 15, there must be a corresponding subgroup that contributes additional elements. The statement does not hold universally as the presence of one or more groups can lead to overlaps in their element counts. Therefore, it is not necessarily true that having more than 8 elements of order 15 implies having at least 16 elements of order 15, making Statement 2 also **False**. Conclusion: Both statements are false: - Statement 1: False

田径运动是一项古老而受人喜爱的体育运动，不仅仅是一种竞技项目，更是一种锻炼身心的方式。通过参与田径运动健身，不仅能够锻炼肌肉，提高心肺功能，还可以拓展人际关系，培养自律意识。

强身健体，塑造完美体态

田径运动包括各种跑步、跳远、投掷等项目，这些动作可以全面锻炼肌肉群，促进新陈代谢，帮助排毒减肥。长期坚持田径锻炼，能够塑造出让人羡慕的完美体态，让您焕发健康活力。

挑战自我，超越极限

参与田径运动，每一次训练和比赛都是对自己极限的挑战。通过不断努力和奋斗，您可以突破自身的心理和身体极限，实现个人发展的突破和成长。

健康身心，提升自信

田径运动的锻炼不仅有助于身体健康，还能够提升心理健康水平。锻炼过程中释放的内啡肽和多巴胺等神经递质可以促进愉悦感，缓解压力，提升自信心。

创造更多可能，获得更多机会

通过田径运动健身，您将结识更多志同道合的朋友，拓展社交圈子，塑造积极向上的生活态度。这也会为您带来更多职业发展的机会和成就感。

结语

田径运动健身的多重价值涵义不仅仅局限于体育竞技，更是一种生活方式，一种健康、积极、快乐的生活方式。让我们一起加入田径运动的行列，享受健康、快乐的人生！